1
1.
Chapter 1
Objectives and Methods of Solid Mechanics
1.1. Defining a problem in solid mechanics
1.1.1.




For each of the following applications, outline briefly:
What would you calculate if you were asked to model the component for a design application?
What level of detail is required in modeling the geometry of the solid?
How would you model loading applied to the solid?
Would you conduct a static or dynamic analysis? Is it necessary to account for thermal stresses? Is
it necessary to account for temperature variation as a function of time?
 What constitutive law would you use to model the material behavior?
1.1.1.1.
A load cell intended to model forces applied to a specimen in a tensile testing machine
1.1.1.2.
The seat-belt assembly in a vehicle
1.1.1.3.
The solar panels on a communications satellite.
1.1.1.4.
A compressor blade in a gas turbine engine
1.1.1.5.
A MEMS optical switch
1.1.1.6.
An artificial knee joint
1.1.1.7.
A solder joint on a printed circuit board
1.1.1.8.
An entire printed circuit board assembly
1.1.1.9.
The metal interconnects inside a microelectronic circuit
1.1.2. What is the difference between a linear elastic stress-strain law and a hyperelastic stress-strain law?
Give examples of representative applications for both material models.
1.1.3. What is the difference between a rate-dependent (viscoplastic) and rate independent plastic
constitutive law? Give examples of representative applications for both material models.
1.1.4. Choose a recent publication describing an application of theoretical or computational solid
mechanics from one of the following journals: Journal of the Mechanics and Physics of Solids;
International Journal of Solids and Structures; Modeling and Simulation in Materials Science and
Engineering; European Journal of Mechanics A; Computer methods in Applied Mechanics and
Engineering. Write a short summary of the paper stating: (i) the goal of the paper; (ii) the problem that
was solved, including idealizations and assumptions involved in the analysis; (iii) the method of
analysis; and (iv) the main results; and (v) the conclusions of the study
2
Chapter 2
2.
Governing Equations
2.1. Mathematical Description of Shape Changes in Solids
e
2
2.1.1. A thin film of material is deformed in simple shear during a plate
impact experiment, as shown in the figure.
Undeformed
h
2.1.1.1.
Write down expressions for the displacement field in the
e1
film, in terms of x1 , x2 d and h, expressing your answer as
e2
components in the basis shown.
d
2.1.1.2.
Calculate the Lagrange strain tensor associated with the
Deformed
deformation, expressing your answer as components in the h
basis shown.
e1
2.1.1.3.
Calculate the infinitesimal strain tensor for the deformation,
expressing your answer as components in the basis shown.
2.1.1.4.
Find the principal values of the infinitesimal strain tensor, in terms of d and h
2.1.2.
Find a displacement field corresponding to a uniform infinitesimal strain field  ij . (Don’t make
this hard – in particular do not use the complicated approach described in Section 2.1.20. Instead, think
about what kind of function, when differentiated, gives a constant). Is the displacement unique?
2.1.3.
Find a formula for the displacement field that generates zero infinitesimal strain.
2.1.4.
Find a displacement field that corresponds to a uniform Lagrange strain tensor Eij .
Is the
displacement unique? Find a formula for the most general displacement field that generates a uniform
Lagrange strain.
2.1.5. The displacement field in a homogeneous, isotropic
circular shaft twisted through angle  at one end is
e2
given by
e1
x 
x 
u1  x1[cos  3   1]  x2 sin  3 

e3
 L 
 L 
x 
x 
u2  x1 sin  3   x2 [cos  3   1]
L
 L 
 L 
u3  0
2.1.5.1.
Calculate the matrix of components of the deformation gradient tensor
2.1.5.2.
Calculate the matrix of components of the Lagrange strain tensor. Is the strain tensor a
function of x3 ? Why?
2.1.5.3.
Find an expression for the increase in length of a material fiber of initial length dl, which is
on the outer surface of the cylinder and initially oriented in the e 3 direction.
3
2.1.5.4.
2.1.5.5.
2.1.5.6.
2.1.5.7.
Show that material fibers initially oriented in the e 2 and e 2 directions do not change their
length.
Calculate the principal values and directions of the Lagrange strain tensor at the point
x1 a , x2 0, x3 0 . Hence, deduce the orientations of the material fibers that have
the greatest and smallest increase in length.
Calculate the components of the infinitesimal strain tensor. Show that, for small values of
 , the infinitesimal strain tensor is identical to the Lagrange strain tensor, but for finite
rotations the two measures of deformation differ.
Use the infinitesimal strain tensor to obtain estimates for the lengths of material fibers
initially oriented with the three basis vectors. Where is the error in this estimate greatest?
How large can  be before the error in this estimate reaches 10%?
2.1.6. To measure the in-plane deformation of a
e2
e2
(c)
sheet of metal during a forming process, your
managers place three small hardness indentations
(c)
2cm
on the sheet. Using a travelling microscope, they
2.8cm
1.414cm
determine that the initial lengths of the sides of
1cm
the triangle formed by the three indents are 1cm,
(a)
(b)
e1
(a)
e1
1cm, 1.414cm, as shown in the picture below.
1.5cm
After deformation, the sides have lengths 1.5cm,
(b)
1cm
2.0cm and 2.8cm. Your managers would like to
use this information to determine the in—plane components of the Lagrange strain tensor.
Unfortunately, being business economics graduates, they are unable to do this.
2.1.6.1.
Explain how the measurements can be used to determine E11, E22 , E12 and do the
calculation.
2.1.6.2.
Is it possible to determine the deformation gradient from the measurements provided?
Why? If not, what additional measurements would be required to determine the
deformation gradient?
2.1.7. To track the deformation in a slowly moving glacier,
three survey stations are installed in the shape of an
equilateral triangle, spaced 100m apart, as shown in the
picture. After a suitable period of time, the spacing
between the three stations is measured again, and found
to be 90m, 110m and 120m, as shown in the figure.
Assuming that the deformation of the glacier is
homogeneous over the region spanned by the survey
stations, please compute the components of the Lagrange
strain tensor associated with this deformation, expressing
your answer as components in the basis shown.
e2
100m
e2
100m
e1
100m
Before deformation
120m
110m
e1
90m
After deformation
2.1.8. Compose a limerick that will help you to remember the distinction between engineering shear
strains and the formal (mathematical) definition of shear strain.
2.1.9. A rigid body motion is a nonzero displacement field that does not distort any infinitesimal volume
element within a solid. Thus, a rigid body displacement induces no strain, and hence no stress, in the
solid. The deformation corresponding to a 3D rigid rotation about an axis through the origin is
y  R  x or yi  Rij x j
where R must satisfy R  RT  RT  R  I , det(R)>0.
4
2.1.9.1.
2.1.9.2.
2.1.9.3.
Show that the Lagrange strain associated with this deformation is zero.
As a specific example, consider the deformation
 y1  cos  sin  0   x1 
 y    sin  cos  0   x 
 2 
 2
 y3   0
0
1   x3 
This is the displacement field caused by rotating a solid through an angle  about the e3
axis. Find the deformation gradient for this displacement field, and show that the
deformation gradient tensor is orthogonal, as predicted above. Show also that the
infinitesimal strain tensor for this displacement field is not generally zero, but is of order
 2 if  is small.
If the displacements are small, we can find a simpler representation for a rigid body
displacement. Consider a deformation of the form
yi ijk  j xk
Here ω is a vector with magnitude <<1, which represents an infinitesimal rotation about an
axis parallel to ω . Show that the infinitesimal strain tensor associated with this
displacement is always zero. Show further that the Lagrange strain associated with this
displacement field is
1
Eij  ijk k  i j
2
This is not, in general, zero. It is small if all k  1 .


2.1.10. The formula for the deformation due to a rotation through an angle  about an axis parallel to a
unit vector n that passes through the origin is
yi  cos ij  (1  cos )ni n j  sin  ikj nk  x j
2.1.10.1.
2.1.10.2.
Calculate the components of corresponding deformation gradient
Verify that the deformation gradient satisfies Fik F jk  Fki Fkj  ij
2.1.10.3.
2.1.10.4.
Find the components of the inverse of the deformation gradient
Verify that both the Lagrange strain tensor and the Eulerian strain tensor are zero for this
deformation. What does this tell you about the distorsion of the material?
Calculate the Jacobian of the deformation gradient. What does this tell you about volume
changes associated with the deformation?
2.1.10.5.
2.1.11. In Section 2.1.6 it was stated that the Eulerian
strain tensor Eij* can be used to relate the length
n
of a material fiber in a deformable solid before
and after deformation, using the formula
2
l  l0
2
2
 Eij* ni n j
2l
where ni are the components of a unit vector
parallel to the material fiber after deformation.
Derive this result.
l
l0
e2
e1
e3
Original
Configuration
Deformed
Configuration
5
2.1.12. Suppose that you have measured the Lagrange strain tensor for a deformation, and wish to calculate
the Eulerian strain tensor. On purely physical grounds, do you think this is possible, without calculating
the deformation gradient? If so, find a formula relating Lagrange strain Eij to Eulerian strain Eij* .
2.1.13. Repeat problem 2.1.6, but instead of calculating the Lagrange strain tensor, find the components of
the Eulerian strain tensor Eij* (you can do this directly, or use the results of problem 2.1.12, or both)
2.1.14. Repeat problem 2.1.7, but instead of calculating the Lagrange strain tensor, find the components of
the Eulerian strain tensor Eij* (you can do this directly, or use the results for problem 2.1.12, or both)
2.1.15. The Lagrange strain tensor can be used to calculate the change in angle between any two material
fibers in a solid as the solid is deformed. In this problem you will calculate the formula that can be used
to do this. To this end, consider two infinitesimal material fibers in the undeformed solid, which are
characterized by vectors with components dxi(1)  l1mi(1) and dxi(2)  l2 mi(2) , where m (1) and m (2) are
two unit vectors. Recall that the angle 0 between
calculated from cos0  mi(1) mi(2) .
m (1) and m (2) before deformation can be
Let dyi(1) and dyi(2) represent the two material fibers after
deformation. Show that the angle between dyi(1) and dyi(2) can be calculated from the formula
cos1 
2 Eij mi(1) m(2)
j  cos  0
1  2 Eij mi(1) m(1)
1  2 Eij mi(2) m(2)
j
j
F
R
dx
e2
x
e1
e3
Original
Configuration
y
dz
dy
z
After first
deformation
After second
deformation
2.1.16. Suppose that a solid is subjected to a sequence of two homogeneous deformations (i) a rigid
rotation R, followed by (ii) an arbitrary homogeneous deformation F. Taking the original configuration
as reference, find formulas for the following deformation measures for the final configuration of the
solid, in terms of F and R:
2.1.16.1. The deformation gradient
2.1.16.2. The Left and Right Cauchy-Green deformation tensors
2.1.16.3. The Lagrange strain
2.1.16.4. The Eulerian strain.
2.1.17. Repeat problem 2.1.16, but this time assume that the sequence of the two deformations is reversed,
i.e. the solid is first subjected to an arbitrary homogeneous deformation F, and is subsequently
subjected to a rigid rotation R.
6
2.1.18. A spherical shell (see the figure) is made from an
incompressible material. In its undeformed state, the inner and
outer radii of the shell are A, B . After deformation, the new
values are a, b . The deformation in the shell can be described (in
Cartesian components) by the equation
1/ 3 x
i
yi  R3  a3  A3
R  xk xk
R
2.1.18.1.
Calculate the components of the deformation gradient
tensor
2.1.18.2. Verify that the deformation is volume preserving
2.1.18.3. Find the deformed length of an infinitesimal radial
line that has initial length l0 , expressed as a function of R
2.1.18.4. Find the deformed length of an infinitesimal circumferential line that has initial length l0 ,
expressed as a function of R
2.1.18.5. Using the results of 2.1.18.3, 2.1.18.4, find the principal stretches for the deformation.
2.1.18.6. Find the inverse of the deformation gradient, expressed as a function of yi . It is best to do
this by working out a formula that enables you to calculate xi in terms of yi and

r

yi yi and differentiate the result rather than to attempt to invert the result of 10.1.
2.1.19. Suppose that the spherical shell described in Problem 2.1.18 is continuously expanding (visualize a
balloon being inflated). The rate of expansion can be characterized by the velocity va  da / dt of the
surface that lies at R=A in the undeformed cylinder.
2.1.19.1. Calculate the velocity field vi  dyi / dt in the sphere as a function of xi
2.1.19.2. Calculate the velocity field as a function of yi (there is a long, obvious way to do this and a
quick, subtle way)
2.1.19.3. Calculate the time derivative of the deformation gradient tensor calculated in 2.1.18.1.
v
2.1.19.4. Calculate the components of the velocity gradient Lij  i by differentiating the result of
y j
2.1.19.5.
2.1.19.6.
2.1.19.1
Calculate the components of the velocity gradient using the results of 2.1.19.3 and 2.1.18.6
Calculate the stretch rate tensor Dij . Verify that the result represents a volume preserving
stretch rate field.
2.1.20. Repeat Problem 2.1.18.1, 2.1.18.6 and all of 2.1.19, but this
time solve the problem using spherical-polar coordinates, using the
various formulas for vector and tensor operations given in
Appendix E. In this case, you may assume that a point with
position x  Re R in the undeformed solid has position vector

y  R3  a3  A3
after deformation.

1/ 3
eR
7
2.1.21. An initially straight beam is bent into a circle with radius R as
e2
shown in the figure. Material fibers that are perpendicular to the
e1
axis of the undeformed beam are assumed to remain perpendicular
to the axis after deformation, and the beam’s thickness and the
length of its axis are assumed to be unchanged. Under these
conditions the deformation can be described as
y1   R  x2  sin( x1 / R)
y2  R  ( R  x2 )cos( x1 / R)
R
m2
m1
where, as usual x is the position of a material particle in the
e2
undeformed beam, and y is the position of the same particle after
e1
deformation.
2.1.21.1. Calculate the deformation gradient field in the beam,
expressing your answer as a function of x1 , x2 , and as
components in the basis {e1, e2 , e3} shown.
2.1.21.2. Calculate the Lagrange strain field in the beam.
2.1.21.3. Calculate the infinitesimal strain field in the beam.
2.1.21.4. Compare the values of Lagrange strain and infinitesimal strain for two points that lie at
( x1  0, x2  h) and ( x1  L, x2  0) . Explain briefly the physical origin of the difference
between the two strain measures at each point. Recommend maximum allowable values of
h/R and L/R for use of the infinitesimal strain measure in modeling beam deflections.
2.1.21.5. Calculate the deformed length of an infinitesimal material fiber that has length l0 and
orientation e1 in the undeformed beam. Express your answer as a function of x2 .
2.1.21.6. Calculate the change in length of an infinitesimal material fiber that has length l0 and
orientation e 2 in the undeformed beam.
2.1.21.7. Show that the two material fibers described in 2.1.21.5 and 2.1.21.6 remain mutually
perpendicular after deformation. Is this true for all material fibers that are mutually
perpendicular in the undeformed solid?
2.1.21.8. Find the components in the basis {e1, e2 , e3} of the Left and Right stretch tensors U and V
as well as the rotation tensor R for this deformation. You should be able to write down U
and R by inspection, without needing to wade through the laborious general process
outlined in Section 2.1.13. The results can then be used to calculate V .
2.1.21.9. Find the principal directions of U as well as the principal stretches. You should be able to
write these down using your physical intuition without doing any tedious calculations.
2.1.21.10. Let {m1 , m 2 , m3} be a basis in which m1 is parallel to the axis of the deformed beam, as
shown in the figure. Write down the components of each of the unit vectors mi in the
basis {e1, e2 , e3} . Hence, compute the transformation matrix Qij  mi  e j that is used to
transform tensor components from {e1, e2 , e3} to {m1 , m 2 , m3} .
2.1.21.11. Find the components of the deformation gradient tensor, Lagrange strain tensor, as well as
U V and R in the basis {m1 , m 2 , m3} .
2.1.21.12. Find the principal directions of V expressed as components in the basis {m1 , m 2 , m3} .
Again, you should be able to simply write down this result.
8
2.1.22. A sheet of material is subjected to a two dimensional
homogeneous deformation of the form
y1  A11x1  A12 x2
y2  A21x1  A22 x2
where Aij are constants. Suppose that a circle of unit radius
e2
m2
m1

e1
is drawn on the undeformed sheet. This circle is distorted
to a smooth curve on the deformed sheet. Show that the
distorted circle is an ellipse, with semi-axes that are parallel
to the principal directions of the left stretch tensor V, and that the lengths of the semi-axes of the ellipse
are equal to the principal stretches for the deformation. There are many different ways to approach this
calculation – some are very involved. The simplest way is probably to assume that the principal
directions of V subtend an angle 0 to the {e1 , e 2 } basis as shown in the figure, write the polar
decomposition A  V  R in terms of principal stretches 1 , 2 and 0 , and then show that y  V  R  x
(where x is on the unit circle) describes an ellipse.
2.1.23. A solid is subjected to a rigid rotation so that a unit vector a in the undeformed solid is rotated to a
new orientation b. Find a rotation tensor R that is consistent with this deformation, in terms of the
components of a and b. Is the rotation tensor unique? If not, find the most general formula for the
rotation tensor.
2.1.24. In a plate impact experiment, a thin film of material with thickness h is
subjected to a homogeneous shear deformation by displacing the upper surface e2
of the film horizontally with a speed v.
2.1.24.1. Write down the velocity field in the film
h
2.1.24.2. Calculate the velocity gradient, the stretch rate and the spin rate
2.1.24.3. Calculate the instantaneous angular velocity of a material fiber
parallel to the e 2 direction in the film
2.1.24.4.
2.1.24.5.
2.1.24.6.
v
Undeformed
e1
Calculate the instantaneous angular velocity of a material fiber parallel to (e1  e2 ) / 2
Calculate the stretch rates for the material fibers in 22.3 and 22.4
What is the direction of the material fiber with the greatest angular velocity? What is the
direction of the material fiber with the greatest stretch rate?
2.1.25. The velocity field v due to a rigid rotation about an axis through the origin can be characterized by
a skew tensor W or an angular velocity vector ω defined so that
v  Wx
v  ω x
W
Find a formula relating the components of
and ω . (One way to approach this problem is to
calculate a formula for W by taking the time derivative of Rodriguez formula – see Sect 2.1.1).
2.1.26. A single crystal deforms by shearing on a single active slip Undeformed
Deformed
system as illustrated in the figure. The crystal is loaded so that
m
tan-1 
the slip direction s and normal to the slip plane m maintain a m
constant direction during the deformation
s
2.1.26.1.
Show that the deformation gradient can be
expressed in terms of the components of the slip
direction s and the normal to the slip plane m as
Fij  ij   si m j where  denotes the shear, as illustrated in the figure.
2.1.26.2.
s
Suppose shearing proceeds at some rate  . At the instant when   0 , calculate (i) the
velocity gradient tensor; (ii) the stretch rate tensor and (iii) the spin tensor associated with
the deformation.
9
2.1.27. The properties of many rubbers and foams are specified by functions of the following invariants of
the left Cauchy-Green deformation tensor Bij  Fik F jk .
I1  trace(B)  Bkk
I2 

 
1 2
1
I1  B B  I12  Bik Bki
2
2

I3  det B  J 2
Invariants of a tensor are defined in Appendix B – they are functions of the components of a tensor that
are independent of the choice of basis.
2.1.27.1. Verify that I1 , I 2 , I 3 are invariants. The simplest way to do this is to show that I1 , I 2 , I 3
are unchanged during a change of basis.
2.1.27.2. In order to calculate stress-strain relations for these materials, it is necessary to evaluate
derivatives of the invariants. Show that
I
I1
I
 2 Fij , 2  2 I1Fij  Bik Fkj , 3  2 I 3 F ji1
Fij
Fij
Fij


2.1.28. The infinitesimal strain field in a long cylinder containing a hole at its center is given by
 31  bx2 / r 2
2.1.28.1.
2.1.28.2.
 32  bx1 / r 2
r  x12  x22
Show that the strain field satisfies the equations of compatibility.
Show that the strain field is consistent with a displacement field of the form u3   , where
  2b tan 1 x2 / x1 . Note that although the strain field is compatible, the displacement
field is multiple valued – i.e. the displacements are not equal at   2 and   0 , which
supposedly represent the same point in the solid. Surprisingly, displacement fields like this
do exist in solids – they are caused by dislocations in a crystal. These are discussed in more
detail in Sections 5.3.4
2.1.29. The figure shows a test designed to
measure the response of a polymer to
large shear strains. The sample is a
hollow cylinder with internal radius a0
e
eR
e er
ez

and external radius a1 . The inside
eR
P eZ
diameter is bonded to a fixed rigid
cylinder.
The external diameter is
Undeformed
bonded inside a rigid tube, which is
Deformed
rotated through an angle  (t ) . Assume
that the specimen deforms as indicated
in the figure, i.e. (a) cylindrical sections
remain cylindrical; (b) no point in the specimen moves in the axial or radial directions; (c) that a
cylindrical element of material at radius R rotates through angle  ( R, t ) about the axis of the
specimen. Take the undeformed configuration as reference. Let ( R, , Z ) denote the cylindrical-polar
coordinates of a material point in the reference configuration, and let {e R , e , e Z } be cylindrical-polar
10
basis vectors at ( R, , Z ) . Let (r , , z ) denote the coordinates of this point in the deformed
configuration, and let {e r , e , e z } by cylindrical-polar basis vectors located at (r ,  , z ) .
2.1.29.6.
2.1.29.7.
2.1.29.8.
2.1.29.9.
(r ,  , z ) in terms of ( R, , Z ) (this constitutes the
Write down expressions for
deformation mapping)
Let P denote the material point at at ( R, , Z ) in the reference configuration. Write down
the reference position vector X of P, expressing your answer as components in the basis
{e R , e , e Z } .
Write down the deformed position vector x of P, expressing your answer in terms of
( R, , Z ) and basis vectors {e R , e , e Z } .
Find the components of the deformation gradient tensor F in {e R , e , e Z } . (Recall that the

1 

gradient operator in cylindrical-polar coordinates is   (e R
 e
 eZ
) ; recall
R
R 
Z
e
e
also that R  e ;   e R )


Show that the deformation gradient can be decomposed into a sequence F  R  S of a
simple shear S followed by a rigid rotation through angle  about the e Z direction R. In
this case the simple shear deformation will have the form
S  e R e R  ee  e Z e Z  kee R
where k is to be determined.
Find the components of F in {e r , e , e z } .
Verify that the deformation is volume preserving (i.e. check the value of J=det(F))
Find the components of the right Cauchy-Green deformation tensors in {e R , e , e Z }
Find the components of the left Cauchy-Green deformation tensor in {e r , e , e z }
2.1.29.10.
2.1.29.11.
2.1.29.12.
2.1.29.13.
Find F 1 in {e R , e , e Z } .
Find the principal values of the stretch tensor U
Write down the velocity field v in terms of (r , , z ) in the basis {e r , e , e z }
Calculate the spatial velocity gradient L in the basis {e r , e , e z }
2.1.29.1.
2.1.29.2.
2.1.29.3.
2.1.29.4.
2.1.29.5.
11
2.2. Mathematical Description of Internal Forces in Solids
2.2.1. A rectangular bar is loaded in a state of uniaxial tension, as shown in the
figure.
2.2.1.1.
Write down the components of the stress tensor in the bar, using
the basis vectors shown.
2.2.1.2.
Calculate the components of the normal vector to the plane
ABCD shown, and hence deduce the components of the traction
vector acting on this plane, expressing your answer as
components in the basis shown, in terms of 
2.2.1.3.
Compute the normal and tangential tractions acting on the plane
shown.
2.2.2.
B
A

D C
e2
e3
e1
Consider a state of hydrostatic stress  ij p ij . Show that the traction vector acting on any
internal plane in the solid (or, more likely, fluid!) has magnitude p and direction normal to the plane.
2.2.3. A cylinder of radius R is partially immersed in a static fluid.
2.2.3.1.
Recall that the pressure at a depth d in a fluid has magnitude
gd . Write down an expression for the horizontal and
vertical components of traction acting on the surface of the
cylinder in terms of  .
2.2.3.2.
Hence compute the resultant force exerted by the fluid on
the cylinder.
2.2.4. The figure shows two designs for a glue joint. The glue
will fail if the stress acting normal to the joint exceeds 60
MPa, or if the shear stress acting parallel to the plane of the
joint exceeds 300 MPa.
2.2.4.1.
Calculate the normal and shear stress acting on
each joint, in terms of the applied stress 
2.2.4.2.
Hence, calculate the value of  that will cause
each joint to fail.
2.2.5.
R
h 
30 0
30 0


Glue joint

For the Cauchy stress tensor with components
100 250 0 
 250 200 0 


 0
0 300 
compute
2.2.5.1.
The traction vector acting on an internal material plane with normal n  (e1  e2 ) / 2
2.2.5.2.
The principal stresses
2.2.5.3.
The hydrostatic stress
2.2.5.4.
The deviatoric stress tensor
2.2.5.5.
The Von-Mises equivalent stress

12
Show that the hydrostatic stress  kk is invariant under a change of basis – i.e. if  ije and  ijm
denote the components of stress in bases {e1, e2 , e3} and {m1 , m 2 , m3} , respectively, show that
2.2.6.
e
m
 kk
  kk
.
2.2.7. A rigid, cubic solid is immersed in a fluid with mass density  .
Recall that a stationary fluid exerts a compressive pressure of
H
magnitude  gh at depth h.
2.2.7.1.
Write down expressions for the traction vector exerted
by the fluid on each face of the cube. You might find it
2a
convenient to take the origin for your coordinate system
at the center of the cube, and take basis vectors
{e1, e2 , e3} perpendicular to the cube faces.
2.2.7.2. Calculate the resultant force due to the tractions acting on the cube, and show that the
vertical force is equal and opposite to the weight of fluid displaced by the cube.
2.2.8. Show that the result of problem 2.2.7 applies to any arbitrarily shaped solid immersed below the
surface of a fluid, i.e. prove that the resultant force acting on an immersed solid with volume V is
Pi   gV  i3 , where it is assumed that e3 is vertical. To do this
2.2.8.1.
Let n j denote the components of a unit vector normal to the surface of the immersed solid
2.2.8.2.
2.2.8.3.
Write down a formula for the traction (as a vector) exerted by the fluid on the immersed
solid
Integrate the traction to calculate the resultant force, and manipulate the result obtain the
required formula.
2.2.9. A component contains a feature with a 90 degree corner as shown in the picture.
The surfaces that meet at the corner are not subjected to any loading. List all the
stress components that must be zero at the corner
2.2.10. In this problem we consider further the beam bending
calculation discussed in Problem 2.1.21. Suppose that the beam
is made from a material in which the Material Stress tensor is
related to the Lagrange strain tensor by
ij  2 Eij
e2
e1
e2
e1
(this can be regarded as representing an elastic material with zero
Poisson’s ratio and shear modulus  )
m2 m1
R
2.2.10.1. Calculate the distribution of material stress in the
e2
bar, expressing your answer as components in the
{e1, e2 , e3} basis
e1
2.2.10.2. Calculate the distribution of nominal stress in the bar
expressing your answer as components in the
{e1, e2 , e3} basis
2.2.10.3. Calculate the distribution of Cauchy stress in the bar expressing your answer as components
in the {e1, e2 , e3} basis
2.2.10.4. Repeat 15.1-15.3 but express the stresses as components in the {m1 , m 2 , m3} basis
13
2.2.10.5.
2.2.10.6.
2.2.10.7.
Calculate the distribution of traction on a surface in the beam that has normal e1 in the
undeformed beam. Give expressions for the tractions in both {e1, e2 , e3} and {m1 , m 2 , m3}
Show that the surfaces of the beam that have positions x2   h / 2 in the undeformed beam
are traction free after deformation
Calculate the resultant moment acting on the ends of the beam.
2.2.11. A solid is subjected to some loading that induces a Cauchy stress  ij(0) at some point in the solid.
The solid and the loading frame are then rotated together so that the entire solid (as well as the loading
frame) is subjected to a rigid rotation Rij . This causes the components of the Cauchy stress tensor to
change to new values  ij(1) . The goal of this problem is to calculate a formula relating  ij(0) ,  ij(1) and
Rij .
2.2.11.1.
Let ni(0) be a unit vector normal to an internal material plane in the solid before rotation.
After rotation, this vector (which rotates with the solid) is ni(1) . Write down the formula
relating ni(0) and ni(1)
2.2.11.2.
Let Ti(0) be the internal traction vector that acts on a material plane with normal ni(0) in the
solid before application of the rigid rotation. Let Ti(1) be the traction acting on the same
material plane after rotation. Write down the formula relating Ti(0) and Ti(1)
2.2.11.3.
Finally, using the definition of Cauchy stress, find the relationship between  ij(0) ,  ij(1) and
Rij .
2.2.12. Repeat problem 2.2.11, but instead, calculate a relationship between the components of Nominal
stress Sij(0) and Sij(1) before and after the rigid rotation.
2.2.13. Repeat problem 2.2.11, but instead, calculate a relationship between the components of material
stress ij(0) and ij(1) before and after the rigid rotation.
14
2.3. Equations of motion and equilibrium for deformable solids
2.3.1. A prismatic concrete column of mass density  supports its own
weight, as shown in the figure. (Assume that the solid is subjected to
a uniform gravitational body force of magnitude g per unit mass).
2.3.1.1.
Show that the stress distribution
 22    g ( H  x2 )
H
e2
satisfies the equations of static equilibrium
 ij
 b j  0
xi
e3
and also satisfies the boundary conditions  ij ni  0
x2
e1
on all free boundaries.
2.3.1.2.
Show that the traction vector acting on a plane with
normal n  sin  e1  cos e2 at a height x2 is given by
T    g ( H  x2 ) cos  e2
2.3.1.3.
Deduce that the normal component of traction acting on the plane is
Tn    g ( H  x2 )cos 2 
2.3.1.4.
show also that the tangential component of traction acting on the plane is
Tt   g ( H  x2 )sin  cos (cos e1  sin  e2 )
(the easiest way to do this is to note that T  Tnn  Tt and solve for the tangential traction).
2.3.1.5.
Suppose that the concrete contains a large number of randomly oriented microcracks. A
crack which lies at an angle  to the horizontal will propagate if
Tt  Tn   0
2.3.1.6.
where  is the friction coefficient between the faces of the crack and  0 is a critical shear
stress that is related to the size of the microcracks and the fracture toughness of the
concrete, and is therefore a material property.
Assume that   1 . Find the orientation of the microcrack that is most likely to propagate.
Hence, find an expression for the maximum possible height of the column.
2.3.2. Is the stress field given below in static equilibrium? If not, find the acceleration or body force
density required to satisfy linear momentum balance
11  Cx1x2
12   21  C (a 2  x22 )
 33   23  13  0
2.3.3. Let  be a twice differentiable, scalar function of position. Derive a plane stress field from  by
setting
11 
 2
 22 
 2
12   21  
 2
x1x2
x22
x12
Show that this stress field satisfies the equations of stress equilibrium with zero body force.
15
2.3.4.
The stress field
 ij 
3Pk xk xi x j
R  xk xk
4 R5
represents the stress in an infinite, incompressible elastic solid that is subjected to a point force with
components Pk acting at the origin (you can visualize a point force as a very large body force which is
concentrated in a very small region around the origin).
2.3.4.1.
Verify that the stress field is in static equilibrium
2.3.4.2.
Consider a spherical region of material centered at the origin. This region is subjected to
(1) the body force acting at the origin; and (2) a force exerted by the stress field on the outer
surface of the sphere. Calculate the resultant force exerted on the outer surface of the
sphere by the stress, and show that it is equal in magnitude and opposite in direction to the
body force.
2.3.5. In this problem, we consider the internal
forces in the polymer specimen described in
Problem 2.1.29 (you will need to solve 2.1.29
before you can attempt this one). Suppose
that the specimen is homogeneous, has mass
density  in the reference configuration, and
may be idealized as a viscous fluid, in which
the Kirchhoff stress is related to stretch rate
by
e
P
eR
eZ
Undeformed
e er
ez

eR
Deformed
τ   D  pI
where p is an indeterminate hydrostatic
pressure and  is the viscosity.
2.3.5.1.
2.3.5.2.
2.3.5.3.
2.3.5.4.
2.3.5.5.
2.3.5.6.
Find expressions for the Cauchy stress tensor, expressing your answer as components in
{e r , e , e z }
Assume steady, quasi-static deformation (neglect accelerations). Express the equations of
equilibrium in terms of  ( r , t )
Solve the equilibrium equation, together with appropriate boundary conditions, to calculate
 (r , t )
Find the torque necessary to rotate the external cylinder
Calculate the acceleration of a material particle in the fluid
Estimate the rotation rate  where inertia begins to play a significant role in determining
the state of stress in the fluid
16
2.4. Work done by stresses; the principle of virtual work
2.4.1. A solid with volume V is subjected to a distribution of traction ti on its surface. Assume that the
solid is in static equilibrium. By considering a virtual velocity of the form  vi  Aij y j , where Aij is a
constant symmetric tensor, use the principle of virtual work to show that the average stress in a solid
can be computed from the shape of the solid and the tractions acting on its surface using the expression
1
1 1
 ij dV   ti y j  t j yi dA

V
V 2

V

S
2.4.2. The figure shows a cantilever beam that is subjected to surface
loading q ( x1 ) per unit length. The state of stress in the beam can be
e2

approximated by 11  M ( x1 ) x2 / I , where I  x22 dA is the area
q(x1)
A
e1
moment of inertia of the beam’s cross section and M ( x1 ) is an arbitrary
function (all other stress components are zero). By considering a virtual velocity field of the form
w( x1 )
 v1  
x2
 v2  w( x1 )
dx1
where w( x1 ) is an arbitrary function satisfying w  0 at x1  0 , show that the beam is in static
equilibrium if
L
L
d 2w
M ( x1 )
dx  q( x1 ) wdx1  0
2 1
dx
1
0
0


By integrating the first integral expression by parts twice, show that the equilibrium equation and
boundary conditions for M ( x1 ) are
d 2M
dx12
 q( x1 )  0
M ( x1 ) 
dM ( x1 )
0
dx1
x1  L
2.4.3. The figure shows a plate with a clamped edge that is subjected to a
pressure p( x1, x2 ) on its surface. The state of stress in the plate can be
approximated by
  M ( x1, x2 ) x3 / 3h3
 33   3  0
where the subscripts  ,  can have values 1 or 2, and M ( x1, x2 ) is
a tensor valued function. By considering a virtual velocity of the form
w
 v  
x3
 v3  w( x1, x2 )
x
where w( x1, x2 ) is an arbitrary function satisfying w  0 on the edge of the plate, show that the beam is
in static equilibrium if
17

M ( x1 )
A
2w
dA  p( x1, x2 )wdA  0
x x

A
By applying the divergence theorem appropriately, show that the governing equation for M ( x1, x2 ) is
 2 M
x x
 p0
2.4.4. The shell shown in the figure is subjected to a radial body
force b  b( R)e R , and a radial pressure pa , pb acting on the
surfaces at R  a and R  b . The loading induces a spherically
symmetric state of stress in the shell, which can be expressed in
terms of its components in a spherical-polar coordinate system as
 RRe R  e R   e  e   e  e .
By considering a
virtual velocity of the form  v  w( R)e R , show that the stress state
is in static equilibrium if
b
b
dw
w

2
2
2
2
   
 RR
 4 R dR  b( R) w( R)4 R dR  4 a pa w(a) 4 b pb w(b)  0
dR
R

a
a




for all w(R). Hence, show that the stress state must satisfy
d RR 1
 2 RR      b  0
 RR   pa  R  a 
dR
R


2.4.5. In this problem, we consider the internal
dissipation in the polymer specimen
described in Problem 2.1.29 and 2.3.5 (you
will need to solve 2.1.29 and 2.3.5 before you
can attempt this one). Suppose that the
specimen is homogeneous, has mass density
 in the reference configuration, and may be
idealized as a viscous fluid, in which the
Kirchhoff stress is related to stretch rate by
e
P
eR
eZ
Undeformed
 RR   pb  R  b 
e er
ez

eR
Deformed
τ   D  pI
where p is an indeterminate hydrostatic
pressure and  is the viscosity.
2.4.5.1.
2.4.5.2.
2.4.5.3.
Calculate the rate of external work done by the torque acting on the rotating exterior
cyclinder
Calculate the rate of internal dissipation in the solid as a function of r.
Show that the total internal dissipation is equal to the external work done on the specimen.
18
3.
Chapter 3
Constitutive Models: Relations between Stress and Strain
3.1. General Requirements for Constitutive Equations
The following problems illustrate the consequences of general restrictions on constitutive equations:
Linear elastic materials: 3.2.6, 3.2.7
Hyperelastic materials: 3.3.2
Generalized Hooke’s law: 3.4.3
Hyperelastic materials: 3.5.8, 3.5.9.
3.2. Linear Elastic Constitutive Equations
3.2.1. Using the table of values given in Section 3.2.4, find values of bulk modulus, Lame modulus, and
shear modulus for steel, aluminum and rubber.
3.2.2. A specimen of an isotropic, linear elastic material with Young’s modulus E and is placed inside a
rigid box that prevents the material from stretching in any direction. This means that the strains in the
specimen are zero. The specimen is then heated to increase its temperature by T . Find a formula for
the stress in the specimen. Find a formula for the strain energy density. How much strain energy would
be stored in a 1cm3 sample of steel if its temperature were increased by 100C? Compare the strain
energy with the heat required to change the temperature by 100C – the specific heat capacity of steel is
about 470 J/(kg-C)
3.2.3. A specimen of an isotropic, linear elastic solid is free of stress, and is heated to increase its
temperature by T . Find expressions for the strain and displacement fields in the solid.
3.2.4. A thin isotropic, linear elastic thin film with Young’s modulus
E, Poisson’s ratio  and thermal expansion coefficient  is bonded
to a stiff substrate. The film is stress free at some initial
temperature, and then heated to increase its temperature by T . The
substrate prevents the film from stretching in its own plane, so that
11   22  12  0 , while the surface is traction free, so that the
film deforms in a state of plane stress. Calculate the stresses in the
film in terms of material properties and temperature, and deduce an
expression for the strain energy density in the film.
e3
e1
19
3.2.5.
A cubic material may be characterized either by its moduli as
0
0   11 
 11  c11 c12 c12 0
  
c11 c12 0
0
0    22 
 22  
 33  
c11 0
0
0    33 




sym
c44 0
0   212 
12  
13  
0 c44 0   213 

 


0
0 c44   2 23 
 23  
or by the engineering constants
0
0
0   11 
 11   1/ E  / E  / E
  


0
0
0   22 
 22    / E 1/ E  / E
  33    / E  / E 1/ E
0
0
0   33 




0
0
1/ 
0
0  12 
 212   0
 213   0
0
0
0 1/ 
0  13 

 


0
0
0
0 1/    23 
 2 23   0
Calculate formulas relating cij to E, and  , and deduce an expression for the anisotropy factor
A  2 (1  ) / E
3.2.6.
Suppose that the stress-strain relation for a linear elastic solid is expressed in matrix form as
   C   , where   ,   and C  represent the stress and strain vectors and the matrix of elastic
constants defined in Section 3.2.8. Show that the material has a positive definite strain energy density
( U  0    0 ) if and only if the eigenvalues of C  are all positive.
3.2.7. Write down an expression for the increment in stress resulting from an increment in strain applied
to a linear elastic material, in terms of the matrix of elastic constants C  . Hence show that, for a linear
elastic material to be stable in the sense of Drucker, the eigenvalues of the matrix of elastic constants
C  must all be positive or zero.
3.2.8. Let   ,   and C  represent the stress and strain vectors and the matrix of elastic constants in
the isotropic linear elastic constitutive equation


0
0
0 
1  
 
1 

0
0
0 

 11 
 11 
1 
 

1 
0
0
0 
 

1 

   22 
 22 
 
1  2 
 0
   33  ET 1 
 33 
E
0
0
0
0





 
2
  2 23  1  2 0 
 23  (1   )(1  2 ) 
1  2 
 0
13 
0
0
0
0
0   213 





 
2
0 

  212 
12 
1  2  

 0
0
0
0
0

2 
20
3.2.8.1.
Calculate the eigenvalues of the stiffness matrix C  for an isotropic solid in terms of
3.2.8.2.
Young’s modulus and Poisson’s ratio. Hence, show that the eigenvalues are positive (a
necessary requirement for the material to be stable – see problem ??) if and only if
1    1/ 2 and E  0 .
Find the eigenvectors of C  and sketch the deformations associated with these
eigenvectors.
3.2.9. Let   ,   and C  represent the stress and strain vectors and the matrix of elastic constants in
the isotropic linear elastic constitutive equation for a cubic crystal
0
0   11 
 11  c11 c12 c12 0
  
c11 c12 0
0
0    22 
 22  
 33  
c11 0
0
0    33 




sym
c44 0
0   2 23 
 23  
12  
0 c44 0   212 

 


0
0 c44   213 
13  
Calculate the eigenvalues of the stiffness matrix C  and hence find expressions for the admissible
ranges of c11, c12 , c44 for the eigenvalues to be positive.
e
3.2.10. Let Cijkl
denote the components of the elasticity tensor in a basis e1, e2 , e3  . Let
m1, m2 , m3
be a second basis, and define ij  mi  e j . Recall that the components of the stress and strain tensor in
e1, e2 , e3
(e)
(e)
and m1, m2 , m3  are related by  ij(m)  ik kl
 jl  ij(m)  ik  kl
 jl . Use this result,
together with the elastic constitutive equation, to show that the components of the elasticity tensor in
m1, m2 , m3 can be calculated from
(m )
e)
Cijkl
 ip  jq C (pqrs
kr ls
3.2.11. Consider a cube-shaped specimen of an anisotropic, linear elastic material. The tensor of elastic
moduli and the thermal expansion coefficient for the solid (expressed as components in an arbitrary
basis) are
e
Cijkl
, ije . The solid is placed inside a rigid box that prevents the material from stretching
in any direction. This means that the strains in the specimen are zero. The specimen is then heated to
increase its temperature by T . Find a formula for the strain energy density, and show that the result
is independent of the orientation of the material with respect to the box.
3.2.12. The figure shows a cubic crystal.
Basis vectors
{e1, e2 , e3} are aligned perpendicular to the faces of the cubic
unit cell. A tensile specimen is cut from the cube – the axis of
the specimen lies in the {e1, e2} plane and is oriented at an
angle  to the e1 direction. The specimen is then loaded in
uniaxial tension  nn parallel to its axis. This means that the
stress components in the basis {n1, n2 , n3} shown in the
picture are
21
 nn
 0

 0
3.2.12.1.
0 0
0 0 
0 0 
Use the basis change formulas for tensors to calculate the components of stress in the
{e1, e2 , e3} basis in terms of  .
Use the stress-strain equations in Section 3.1.16 to find the strain components in the
{e1, e2 , e3} basis, in terms of the engineering constants E , ,  for the cubic crystal. You
need only calculate 11,  22 , 12 .
3.2.12.2.
3.2.12.3.
3.2.12.4.
Use the basis change formulas again to calculate the strain components in the {n1, n2 , n3}
basis oriented with the specimen. Again, you need only calculate 11,  22 , 12 . Check
your answer by setting   E / 2(1  ) - this makes the crystal isotropic, and you should
recover the isotropic solution.
Define the effective axial Young’s modulus of the tensile specimen as E ( )   nn /  nn ,
where  nn  n  ε  n is the strain component parallel to the n1 direction. Find a formula
for E ( ) in terms of E , ,  .
Using data for copper, plot a graph of E ( ) against  . For copper, what is the orientation
that maximizes the longitudinal stiffness of the specimen? Which orientation minimizes
the stiffness?
3.3. Hypoelasticity
3.3.1. A thin-walled tube can be idealized using the hypoelastic constitutive
equation described in Section 3.3. You may assume that the axial load induces a
uniaxial stress  zz  P /(2 at ) while the torque induces a shear stress
 z  Q /(2 a 2t ) . The shear strains are related to the twist per unit length of
the tube by  z   / 2 L , while the axial strains are related to the extension of the
tube by  zz   / L .
3.3.1.1.
Calculate a relationship between the axial load P and the extension
 for a tube subjected only to axial loading
3.3.1.2.
Calculate a relationship between the torque Q and the twist  for a
tube subjected only to torsional loading
3.3.1.3.
Calculate a relationship between P, Q and  ,  for a tube subjected to combined axial and
torsional loading.
3.3.2. Consider a material with the hypoelastic constitutive equation described in Section 3.3. Calculate an
expression for the tangent stiffness Cijkl   ij /  kl . Express your answer in matrix form by finding
the matrix [C ] such that stress increment d   d11, d 22 , d 33 , d 23 , d13 , d12 
T
and strain
increment d    d 11, d  22 , d  33 , d  23 , d 13 , d 12  are related by d  [C ]d  . Find the eigenvalues
T
of [C ] for the particular case [ ]  [11,  22 ,  33 ,0,0,0] . Hence, show that the material is stable in the
sense of Drucker as long as K  0 ,  0  0 .
22
3.4. Generalized Hooke’s law: Materials subjected to small strains and large
rotations
3.4.1. A uniaxial tensile specimen with length L and cross-sectional area A is idealized with a constitutive
law that relates the material stress ij to the Lagrange strain Eij by
E 


Ekk ij 
 Eij 
1  
1  2

where E and  are elastic constants. The specimen is subjected to a uniaxial force P which induces an
extension  . Calculate the relationship between P and  , and compare the results with the
predictions of a linear elastic constitutive equation.
ij 
3.4.2. A thin walled tube with length L, radius a and wall thickness t is subjected to
a torque Q . The tube can be idealized using the constitutive equation described
in the preceding problem. Assume that during deformation plane sections of the
tube remain plane, and that cross sections of the tube rotate through and angle
 ( z)   z / L .
3.4.2.1.
Calculate an expression for the Lagrange strain in the specimen
3.4.2.2.
Hence deduce an expression for the material stress in the tube
3.4.2.3.
Compute the Cauchy stress distribution
3.4.2.4.
Hence, deduce an expression relating the torque Q to the tube’s twist
 . Compare the result with the predictions of a simple linear elastic
constitutive equation.
3.4.3. Check whether the constitutive equation given in problem 3.4.1 satisfies the test for objectivity
described in Section 3.1.
3.5. Hyperelasticity
3.5.1. Derive the stress-strain relations for an
incompressible, Neo-Hookean material subjected
to
3.5.1.1.
Uniaxial tension
3.5.1.2.
Equibiaxial tension
3.5.1.3.
Pure shear
Derive expressions for the Cauchy stress, the
Nominal stress, and the Material stress tensors
(the solutions for nominal stress are listed in the
table in Section 3.5.6). You should use the following procedure: (i) assume that the specimen
experiences the length changes listed in 3.5.6; (ii) use the formulas in Section 3.5.5 to compute the
Cauchy stress, leaving the hydrostatic part of the stress p as an unknown; (iii) Determine the hydrostatic
stress from the boundary conditions (e.g. for uniaxial tensile parallel to e1 you know  22   33  0 ;
for equibiaxial tension or pure shear in the e1 , e 2 plane you know that  33  0 )
23
3.5.2.
Repeat problem 3.5.1 for an incompressible Mooney-Rivlin material.
3.5.3.
Repeat problem 3.5.1 for an incompressible Arruda-Boyce material
3.5.4.
Repeat problem 3.5.1 for an incompressible Ogden material.
3.5.5. Using the results listed in the table in Section 3.5.6, and the material properties listed in Section
3.5.7, plot graphs showing the nominal stress as a function of stretch ratio  for each of (a) a NeoHookean material; (b) a Mooney-Rivlin material; (c) the Arruda-Boyce material and (c) the Ogden
material when subjected to uniaxial tension, biaxial tension, and pure shear (for the latter case, plot the
largest tensile stress S1 ).
3.5.6.
A foam specimen is idealized as an Ogden-Storakers foam with strain energy density

2  
1
U
  2  3  3  ( J   1) 
2 1

 

where  , and  are material properties. Calculate:
3.5.6.1.
The Cauchy stress in a specimen subjected to a pure volume change with principal stretches
1  2  3  
3.5.6.2.
The Cauchy stress in a specimen subjected to volume preserving uniaxial extension
3.5.6.3.
The Cauchy stress in a specimen subjected to uniaxial tension, as a function of the tensile
stretch ratio 1 . (To solve this problem you will need to assume that the solid is subjected
to principal stretches 1 parallel to e1 , and stretches 2 parallel to e 2 and e3 . You will
need to determine 2 from the condition that  22   33  0 in a uniaxial tensile test.
3.5.7.
Suppose that a hyperelastic solid is characterized by a strain energy density U ( I1, I 2 , J ) where
B B 
1
I 2   I12  ik ki  J  det B
2
J
J 4/3 
are invariants of the Left Cauchy-Green deformation tensor Bij  Fik F jk . Suppose that the solid is
subjected to an infinitesimal strain, so that B can be approximated as Bij  ij   ij , where  ij is a
I1 
Bkk
2/3
symmetric infinitesimal strain tensor. Linearize the constitutive equations for  ij  1 , and show that
the relationship between Cauchy stress  ij and infinitesimal strain  ij is equivalent to the isotropic
linear elastic constitutive equation. Give formulas for the bulk modulus and shear modulus for the
equivalent solid in terms of the derivatives of U .
3.5.8. The constitutive law for a hyperelastic solid is derived from a strain energy potential U ( I1 , I 2 , I 3 ) ,
where
1
1
I1  trace(B)  Bkk I 2  I12  B B  I12  Bik Bki I3  det B  J 2
2
2
are the invariants of the Left Cauchy-Green deformation tensor B  F  FT Bij  Fik F jk .

3.5.8.1.
3.5.8.2.
 

Calculate the Cauchy stress induced in the solid when it is subjected to a rigid rotation,
followed by an arbitrary homogeneous deformation.
Hence, demonstrate that the
constitutive law is isotropic.
Apply the simple check described in Section 3.1 to test whether the constitutive law is
objective.
24
3.5.9. The strain energy density of a hyperelastic solid is sometimes specified as a function of the right
Cauchy-Green deformation tensor Cij  Fki Fkj , instead of Bij as described in Section 3.5. (This
procedure must be used if the material is anisotropic, for example)
3.5.9.1.
Suppose that the strain energy density has the general form W (Cij ) . Derive formulas for
the Material stress, Nominal stress and Cauchy stress in the solid as functions of Fij , Cij
and W / Cij
3.5.9.2.
3.5.9.3.
3.5.9.4.
Apply the simple check described in Section 3.1 to demonstrate that the resulting stressstrain relation is objective.
Calculate the Cauchy stress induced in the solid when it is subjected to a rigid rotation,
followed by an arbitrary homogeneous deformation.
Hence, demonstrate that the
constitutive law is not, in general, isotropic.
Suppose that the constitutive law is simplified further by writing the strain energy density
as a function of the invariants of C, i.e. W (Cij )  U ( I1, I 2 , I3 ) , where


1 2
I1  Cik Cki I3  det C  J 2
2
Derive expressions relating the Cauchy stress components to U / I j .
I1  Ckk I 2 
3.5.9.5.
Demonstrate that the simplified constitutive law described in 3.5.9.4 characterizes an
isotropic solid.
3.6. Viscoelasticity

k
3.6.1. The uniaxial tensile stress-strain behavior of a

viscoelastic material is idealized using the spring
k
damper systems illustrated in the figure, as discussed in

Section 3.6.3.


Maxwell
3.6.1.1.
Derive the differential equations relating
stress to strain for each system.
k1
 Kelvin-Voigt
3.6.1.2.
Calculate expressions for the relaxation

modulus for the Maxwell material and the
3 parameter model.
3 parameter
k2
3.6.1.3.
Calculate expressions for the creep
compliance of all three materials

3.6.1.4.
Calculate expressions for the complex
modulus for all three materials.
3.6.1.5.
Calculate expressions for the complex compliance for all three materials.
3.6.2. The shear modulus of a viscoelastic material can be approximated by a Prony series given by
G (t )  G  G1et / t1 .
3.6.2.1.
Find the creep shear compliance of the material
3.6.2.2.
Find the complex shear modulus of the material
3.6.2.3.
Find the complex shear compliance of the material
25
3.6.3. A uniaxial tensile specimen is made from a viscoelastic material with time independent bulk
modulus K, and has a shear modulus that can be approximated by the Prony series
G (t )  G  G1et / t1 . The specimen is subjected to step increase in uniaxial stress, so that 11   0
t  0 with all other stress components zero. Find an expression for the history of strain the specimen.
It is easiest to solve this problem by first calculating the creep shear compliance for the material.
3.6.4. A floor is covered with a pad with thickness h of viscoelastic
material, as shown in the figure. The pad is perfectly bonded to
the floor, so that 11   33  0 . The pad can be idealized as a
viscoelastic solid with time independent bulk modulus K, and has
a shear modulus that can be approximated by the Prony series
e3
h
e1
G (t )  G  G1et / t1 . The surface of the pad is subjected to a
history of displacement u  u (t )e 2 .
3.6.4.1.
Calculate the history of stress induced in the pad by u (t )  0 t  0 u (t )  u0 t  0
3.6.4.2.
Calculate the history of stress induced in the pad by u (t )  0 t  0 u (t )  u0 sin t t  0
3.6.4.3.
Assume that the pad is subjected to a displacement u (t )  u0 sin t for long enough for the
cycles of stress and strain to settle to steady state. Calculate the total energy dissipated per
unit area of the pad during a cycle of loading.
27 0C
Time Modulus
(Sec)
(GPa)
9
23.95
17
23.91
29
23.84
52
23.72
93
23.72
166
23.51
295
23.16
525
22.61
933
21.79
1660
20.73
2952
19.66
5250
18.89
9337
18.46
16600 18.04
35 0C
Time Modulus
(Sec)
(GPa)
13
18.89
23
18.46
41
18.04
73
17.36
131
16.27
233
14.67
415
12.63
738
10.61
1312
9.29
2334
8.81
4151
8.61
7381
8.34
13126 7.91
35 0C
Time Modulus
(Sec)
(GPa)
17
8.81
31
8.61
55
8.34
98
7.91
174
7.27
310
6.46
552
5.67
982
5.16
1746
5.00
3106
4.97
5523
4.95
9822
4.92
17467 4.86
55 0C
Time Modulus
(Sec)
GPa
15
5.67
27
5.16
48
5.00
85
4.97
152
4.95
271
4.92
482
4.86
857
4.76
1524
4.59
2711
4.32
4822
3.92
8574
3.41
15248 2.92
65 0C
Time Modulus
(Sec)
GPa
17
4.92
30
4.86
55
4.76
97
4.59
174
4.32
309
3.92
550
3.41
979
2.92
1742
2.60
3097
2.50
5508
2.50
8000
2.50
17500 2.50
3.6.5. The table above lists measured relaxation (shear) modulus for a (fictitious) polymer at various
temperatures. The polymer has a glass transition temperature of 30 0C.
3.6.5.1.
Plot a graph of the modulus as a function of time for each temperature, using a log scale for
both axes
3.6.5.2.
Hence, show that the data for various temperatures can be collapsed onto a single mastercurve by scaling the times in each experiment by a temperature dependent factor A(T , T1 ) ,
as described in Section 3.6.1. Plot the master-curve corresponding to relaxation at 27 0C.
26
3.6.5.3.
Plot a graph of log( A(T , T1 )) as a function of temperature, and show that the data can be fit
by a function of the form log[ A(T ;T1)]  C1(T  T1) / C2  (T  T1) . Determine the values
of C1 , C2 that best fit the data
3.6.5.4.
3.6.5.5.
Hence, determine the constants C1g , C2g for the material, as discussed in Section 3.6.1
Hence, scale the times in the experimental data to plot the relaxation modulus at the glass
transition temperature G (t , Tg ) .
3.6.5.6.
Find a Prony series fit to G (t , Tg ) .
Use four terms in the series, together with an
appropriate value for G
3.6.6.
An instrument with mass m=10kg is mounted a set of rubber pads with
m
combined cross sectional area A=5 cm2 and height h=3cm as shown in the
figure. The pads are made from polyisobutylene, with properties listed in h
x
Section 3.6.6. The base vibrates harmonically with amplitude Y0 and
y
(angular) frequency  , causing the instrument to vibrate (also harmonically)
with amplitude X 0 .
3.6.6.1.
Find an expression relating X 0 / Y0 to the harmonic modulus G( ) of the material and m,
A, h and  . Assume that the pads are all subjected to a uniaxial state of stress.
3.6.6.2.
Find an expression for the harmonic modulus G( ) in terms of the material properties G
and Gi , ti .
3.6.6.3.
Hence, plot a graph showing the variation of X 0 / Y0 as a function of frequency, for
temperatures 00 C , 250 C ,
400 C
3.7. Small-strain metal plasticity
3.7.1. The stress state induced by stretching a large plate containing a
e2
cylindrical hole of radius a at the origin is given by
  3a 4 a 2 

3a 2
r
11   0 1   4  2  cos 4  2 cos 2 

  2r

r 
2r

 

a
  a 2 3a 4 

e1
a2
 22   0   2  4  cos 4  2 cos 2 
 r

2r 
2r


  3a 4 a 2 

a2
12   0   4  2  sin 4  2 sin 2 
  2r

r 
2r


Here,  0 is the stress in the plate far from the hole. (Stress components not listed are all zero)
3.7.1.1.
Plot contours of von-Mises equivalent stress (normalized by  0 ) as a function of r / a and
 , for a material with   0.3 Hence identify the point in the solid that first reaches yield.
3.7.1.2.
Assume that the material has a yield stress Y .Calculate the critical value of  0 / Y that will
just cause the plate to reach yield.
27
3.7.2. The stress state (expressed in cylindrical-polar coordinates) in a
thin disk with mass density  0 that spins with angular velocity  can
be shown to be
 rr   3  
 
0 2
0 2
8
a2  r 2


(3  )a 2  (3  1)r 2
8
Assume that the disk is made from an elastic-plastic material with yield stress Y and   1/ 3 .
3.7.2.1.
Find a formula for the critical angular velocity that will cause the disk to yield, assuming
Von-Mises yield criterion. Where is the critical point in the disk where plastic flow first
starts?
3.7.2.2.
Find a formula for the critical angular velocity that will cause the disk to yield, using the
Tresca yield criterion. Where is the critical point in the disk where plastic flow first starts?
3.7.2.3.
Using parameters representative of steel, estimate how much kinetic energy can be stored in
a disk with a 0.5m radius and 0.1m thickness.
3.7.2.4.
Recommend the best choice of material for the flywheel in a flywheel energy storage
system.
3.7.3. An isotropic, elastic-perfectly plastic thin film with Young’s
e3
Modulus E , Poisson’s ratio  , yield stress in uniaxial tension Y
and thermal expansion coefficient  is bonded to a stiff
substrate. It is stress free at some initial temperature and then
e1
heated. The substrate prevents the film from stretching in its own
plane, so that 11   22  12  0 , while the surface is traction
free, so that the film deforms in a state of plane stress. Calculate
the critical temperature change Ty that will cause the film to yield, using (a) the Von Mises yield
criterion and (b) the Tresca yield criterion.
3.7.4. Assume that the thin film described in the preceding problem shows so little strain hardening
behavior that it can be idealized as an elastic-perfectly plastic solid, with uniaxial tensile yield stress Y.
Suppose the film is stress free at some initial temperature, and then heated to a temperature Ty ,
where Ty is the yield temperature calculated in the preceding problem, and   1 .
3.7.4.1.
3.7.4.2.
Find the stress in the film at this temperature.
The film is then cooled back to its original temperature. Find the stress in the film after
cooling.
3.7.5. Suppose that the thin film described in the preceding problem is made
from an elastic, isotropically hardening plastic material with a Mises yield
surface, and yield stress-v-plastic strain as shown in the figure. The film
is initially stress free, and then heated to a temperature Ty , where Ty
Y
Y0
h
is the yield temperature calculated in problem 1, and   1 .
3.7.5.1.
Find a formula for the stress in the film at this temperature.
p
3.7.5.2.
The film is then cooled back to its original temperature. Find
the stress in the film after cooling.
3.7.5.3.
The film is cooled further by a temperature change T  0 . Calculate the critical value of
T that will cause the film to reach yield again.
28
3.7.6. Suppose that the thin film described in the preceding problem is made
from an elastic, linear kinematically hardening plastic material with a Mises
yield surface, and yield stress-v-plastic strain as shown in the figure. The
stress is initially stress free, and then heated to a temperature Ty , where

h
Ty is the yield temperature calculated in problem 1, and   1 .
3.7.6.1.
3.7.6.2.
3.7.6.3.

Find a formula for the stress in the film at this temperature.
The film is then cooled back to its original temperature. Find the
Kinematic hardening
stress in the film after cooling.
The film is cooled further by a temperature change T  0 .
Calculate the critical value of T that will cause the film to reach yield again.
3.7.7. A thin-walled tube of mean radius a and wall thickness t<<a is subjected to
an axial load P which exceeds the initial yield load by 10% (i.e. P  1.1PY ). The
axial load is then removed, and a torque Q is applied to the tube. You may
assume that the axial load induces a uniaxial stress  zz  P /(2 at ) while the
torque induces a shear stress  z  Q /(2 a 2t ) . Find the magnitude of Q to
cause further plastic flow, assuming that the solid is
3.7.7.1.
an isotropically hardening solid with a Mises yield surface
3.7.7.2.
a linear kinematically hardening solid with a Mises yield surface
Express your answer in terms of PY and appropriate geometrical terms, and
assume infinitesimal deformation.
3.7.8. A cylindrical, thin-walled pressure vessel with initial radius R, length L and
wall thickness t<<R is subjected to internal pressure p. The vessel is made from
an isotropic elastic-plastic solid with Young’s modulus E, Poisson’s ratio  , and
its yield stress varies with accumulated plastic strain  e as Y  Y0  h e . Recall
that the stresses in a thin-walled pressurized tube are related to the internal
pressure by  zz  pR /(2t ) ,    pR / t
3.7.8.1.
Calculate the critical value of internal pressure required to initiate
yield in the solid
3.7.8.2.
Find a formula for the strain increment d  rr , d  , d  zz resulting from an increment in
pressure dp
3.7.8.3.
Suppose that the pressure is increased 10% above the initial yield value. Find a formula for
the change in radius, length and wall thickness of the vessel. Assume small strains.
3.7.9. Write a simple program that will compute the history of stress resulting from an arbitrary history of
strain applied to an isotropic, elastic-plastic von-Mises solid. Assume that the yield stress is related to
the accumulated effective strain by Y  Y0 (1   p /  0 ) n , where Y0 ,  0 and n are material constants.
Check your code by using it to compute the stress resulting from a volume preserving uniaxial strain
11   ,  22   33   / 2 , and compare the predictions of your code with the analytical solution. Try
one other cycle of strain of your choice.
29
3.7.10. In a classic paper, Taylor, G. I., and Quinney, I., 1932, “The
Plastic Distortion of Metals,” Philos. Trans. R. Soc. London,
Ser. A, 230, pp. 323–362 described a series of experiments
designed to investigate the plastic deformation of various ductile
metals. Among other things, they compared their experimental
measurements with the predictions of the von-Mises and Tresca
yield criteria and their associated flow rules. They used the
apparatus shown in the figure. Thin walled cylindrical tubes
were first subjected to an axial stress  zz   0 . The stress was
sufficient to extend the tubes plastically. The axial stress was
then reduced to a magnitde m 0 , with 0  m  1 , and a
progressively increasing torque was applied to the tube so as to
induce a shear stress  z   in the solid. The twist, extension
and internal volume of the tube were recorded as the torque
was applied. In this problem you will compare their
0.6
experimental results with the predictions of plasticity theory.
Assume that the material is made from an isotropically 
0.4
hardening rigid plastic solid, with a Von Mises yield surface,
and yield stress-v-plastic strain given by Y  Y0  h p .
3.7.10.1. One set of experimental results is illustrated in
the figure to the right. The figure shows the ratio
 /  0 required to initiate yield in the tube during
torsional loading as a function of m. Show that
theory predicts that  /  0 
1  m2  /
0.2
0
Experiment (Cu)
Theory
(isotropic hardening)
0.2
0.4
m
0.6
0.8
1.0
3
Compute the magnitudes of the principal stresses (1, 2 , 3 ) at the point of yielding under
combined axial and torsional loads in terms of  0 and m.
3.7.10.3. Suppose that, for a given axial stress  zz  m 0 , the shear stress  is first brought to the
critical value required to initiate yield in the solid,
0.0
and is then increased by an infinitesimal increment
Experiment
d . Find expressions for the resulting plastic
-0.2
strain increments d  zz , d  , d  rr , d  r , in terms
-0.4
of m,  0 , h and d .
3.7.10.4. Hence, deduce expressions for the magnitudes of
the principal strains increments (d 1 , d  2 , d  3 ) -0.6
Theory
resulting from the stress increment d .
-0.8
(Isotropic hardening)
3.7.10.5. Using the results of 11.2 and 11.6, calculate the
so-called “Lode parameters,” defined as
-1.0 -0.8 -0.6 -0.4 -0.2 0.0
(d   d  3 )
(   3 )
 2 2
1
2 2
1
(d 1  d  3 )
(1   3 )
and show that the theory predicts   
3.7.10.2.

3.7.11. The Taylor/Quinney experiments show that the constitutive equations for an isotropically hardening
Von-Mises solid predict behavior that matches reasonably well with experimental observations, but
there is a clear systematic error between theory and experiment. In this problem, you will compare the
30
predictions of a linear kinematic hardening law with experiment. Assume that the solid has a yield
function and hardening law given by
3
2
f 
(Sij  ij )(Sij  ij )  Y0  0
dij  cd  ijp
2
3
3.7.11.1. Assume that during the initial tensile test, the axial stress  zz   0 in the specimens
reached a magnitude  0   Y0 , where Y0 is the initial tensile yield stress of the solid and
  1 is a scalar multiplier. Assume that the axial stress was then reduced to m 0 and a
progressively increasing shear stress was applied to the solid. Show that the critical value
of  /  0 at which plastic deformation begins is given by
1/ 2
3.7.11.2.
3.7.11.3.
3.7.11.4.
3.7.11.5.
3.7.11.6.

1 
 1

1 2 

 2  (m  1   ) 
0
3



Plot  /  0 against m for various values of  .
Suppose that, for a given axial stress  zz  m 0 , the shear stress is first brought to the
critical value required to initiate yield in the solid, and is then increased by an infinitesimal
increment d . Find expressions for the resulting plastic strain increments, in terms of m,
 , c Y0 and d .
Hence, deduce the magnitudes of the principal strains in the specimen (d 1 , d  2 , d  3 ) .
Compute the magnitudes of the principal stresses (1, 2 , 3 ) at the point of yielding under
combined axial and torsional loads in terms of m,  , and Y0 .
Finally, find expressions for Lode’s parameters
(d   d  3 )
(   3 )
 2 2
1
2 2
1
(d 1  d  3 )
(1   3 )
Plot  versus  for various values of  , and compare your predictions with Taylor and
Quinney’s measurements.
3.7.12. An elastic- nonlinear kinematic hardening solid has Young’s modulus E , Poisson’s ratio  , a
Von-Mises yield surface
3
f ( ij ,ij ) 
Sij  ij Sij  ij  Y  0
2
where Y is the initial yield stress of the solid, and a hardening law given by
2
dij  cd ijp  ij d p
3
where c and  are material properties. In the undeformed solid, ij  0 . Calculate the formulas



relating the total strain increment d  ij to the state of stress  ij , the state variables ij and the
increment in stress d ij applied to the solid
3.7.13. Consider a rigid nonlinear kinematic hardening solid, with yield surface and hardening law
described in the preceding problem.
3.7.13.1. Show that the constitutive law implies that  kk  0
31
3.7.13.2.
3.7.13.3.
Show that under uniaxial loading with 11   ,  22   33  11 / 2
Suppose the material is subjected to a monotonically increasing uniaxial tensile stress
11   .
Show that the uniaxial stress-strain curve has the form
  Y  (c /  )[1  exp( )] (it is simplest to calculate ij as a function of the strain and
then use the yield criterion to find the stress)
3.7.14. Suppose that a solid contains a large number of randomly oriented slip planes, so that it begins to
yield when the resolved shear stress on any plane in the solid reaches a critical magnitude k.
3.7.14.1. Suppose that the material is subjected to principal stresses  1 ,  2 ,  3 . Find a formula for
the maximum resolved shear stress in the solid, and by means of appropriate sketches,
identify the planes that will begin to slip.
3.7.14.2. Draw the yield locus for this material.
3.7.15. Consider a rate independent plastic material with yield criterion f ( ij )  0 . Assume that (i) the
constitutive law for the material has an associated flow rule, so that the plastic strain increment is
related to the yield criterion by d ijp  d p f /  ij ; and (ii) the yield surface is convex, so that
f  ij*   ( ij   ij* )   f  ij*   0


 
for all stress states  ij and  ij* satisfying f ( ij )  0 and f ( ij* )  0 and 0    1 . Show that the
material obeys the principle of maximum plastic resistance.
3.7.16. The yield strength of a frictional material (such as sand) depends on hydrostatic pressure. A simple
model of yield and plastic flow in such a material is proposed as follows:
3
Yield criterion F ( ij )  f ( ij )   kk  0
f ( ij ) 
Sij Sij
2
f
Flow rule d  ijp  d  p
 ij
Where  is a material constant (some measure of the friction between the sand grains).
3.7.16.1. Sketch the yield surface for this material in principal stress space (note that the material
looks like a Mises solid whose yield stress increases with hydrostatic pressure. You will
need to sketch the full 3D surface, not just the projection that is used for pressure
independent surfaces)
3.7.16.2. Sketch a vector indicating the direction of plastic flow for some point on the yield surface
drawn in part (3.8.3.1)
3.7.16.3. By finding a counter-example, demonstrate that this material does not satisfy the principle
of maximum plastic resistance
3.7.16.4.
3.7.16.5.
( ij   ij* )d ijp  0
(you can do this graphically, or by finding two specific stress states that violate the
condition)
Demonstrate that the material is not stable in the sense of Drucker – i.e. find a cycle of
loading for which the work done by the traction increment through the displacement
increment is non-zero.
What modification would be required to the constitutive law to make it satisfy the principle
of maximum plastic resistance and Drucker stability? How does the physical response of
32
the stable material differ from the original model (think about compaction under combined
shear and pressure).
3.8. Viscoplasticity
3.8.1. Suppose that a uniaxial tensile specimen with length made from Aluminum can be characterized by
a viscoplastic constitutive law with properties listed in Section 3.8.4. Plot a graph showing the strain
rate of the specimen as a function of stress. Use log scales for both axes, with a stress range between 5
and 60 MPa, and show data for room temperature; 1000C, 2000C, 3000C, 4000C and 5000C. Would
you trust the predictions of the constitutive equation outside this range of temperature and stress? Give
reasons for your answer.
3.8.2. A uniaxial tensile specimen can be idealized as an elastic-viscoplastic solid, with Young’s modulus
E, and a flow potential given by
m
 
g ( e )   0  e 
 Y 
where Y,  0 and m are material properties. The specimen is stress free at time t=0¸ and is then
stretched at a constant (total) strain rate  .
3.8.2.1.
Show that the equation governing the axial stress in the specimen can be expressed in
d
  ( / Y )( 0 / )1/ m
dimensionless
form
as
where
and
 m 1,
dt
are dimensionless measures of stress and time.
t   Et / Y   0 / 
Hence, deduce that the normalized stress  is a function only of the material parameter m
1/ m
3.8.2.2.
3.8.2.3.
3.8.2.4.
3.8.2.5.
3.8.2.6.
3.8.2.7.
and the normalized strain    ( E / Y )( 0 / )1/ m .
Show that during steady state creep   1 .
Obtain an analytical solution relating  to  for m=1.
Obtain an analytical solution relating  to  for very large m (note that, in this limit the
material behaves like an elastic-perfectly plastic, rate independent solid, with yield stress
Y).
By integrating the governing equation for  numerically, plot graphs relating  to  for a
few values of m between m=1 and m=100.
Estimate the time, and strain, required for a tensile specimen of Aluminum to reach steady
state creep at a temperature of 4000C, when deformed at a strain rate of 10-3s-1
3.8.3. The figure shows a thin polycrystalline film on a substrate. The
film can be idealized as an elastic-viscoplastic solid with uniaxial
strain
rate,
stress
temperature
relation
given
by
   0 exp(Q / kT )  / Y  , where  0 , Q and Y are material
m
constants, and k is the Boltzmann constant.
e3
e1
33
3.8.4. A cylindrical, thin-walled pressure vessel with initial radius R, length L and
wall thickness t<<R is subjected to internal pressure p. The vessel is made from
an elastic-power-law viscoplastic solid with Young’s modulus E, Poisson’s ratio
 , and a flow potential given by
m
 
g ( e )   0  e 
 Y 
where  e is the Von-Mises eequivalent stress. Recall that the stresses in a thinwalled pressurized tube are related to the internal pressure by  zz  pR /(2t ) ,    pR / t . Calculate
the steady-state strain rate in the vessel, as a function of pressure and relevant geometric and material
properties. Hence, calculate an expression for the rate of change of the vessel’s length, radius and wall
thickness as a function of time.
3.9. Large strain, rate dependent plasticity
e2
3.9.1. The figure shows a thin film of material that is deformed plastically
e1
v(y2 )
during a pressure-shear plate impact experiment. The goal of this
problem is to derive the equations governing the velocity and stress
fields in the specimen. Assume that:
 The film deforms in simple shear, and that the velocity v  v( y2 , t )e1 and Kirchoff stress fields
τ  q( y2 , t )(e1  e2  e2  e1 )  (11e1  e1   22e 2  e 2   33e3  e3 ) are independent of x1
 The material has mass density  and isotropic elastic response, with shear modulus  and
Poisson’s ratio 
 The film can be idealized as a finite strain viscoplastic solid with power-law Mises flow potential,
as described in Section 3.9. Assume that the plastic spin is zero.
3.9.1.1.
Calculate the velocity gradient tensor L, the stretch rate tensor D and spin tensor W for the
deformation, expressing your answer as components in the e1 , e 2 basis shown in the figure
3.9.1.2.
Find an expression for the plastic stretch rate, in terms of the stress and material properties

3.9.1.4.
e
Use the elastic stress rate-stretch rate relation  ij  Cijkl Dkl
to obtain an expression for the
time derivative of the shear stress q and the stress components 11 , 22 , 33 in terms of
v ( x2 ) , τ and appropriate material properties
Write down the linear momentum balance equation in terms of τ and v .
3.9.1.5.
How would the governing equations change if W p  W ?
3.9.1.3.
34
3.10. Large Strain Viscoelasticity
3.10.1. A cylindrical specimen is made from a material that can be idealized using the finite-strain
viscoelasticity model described in Section 3.10.
The specimen may be approximated as
incompressible.
3.10.1.1. Let L denote the length of the deformed specimen, and L0 denote the initial length of the
specimen. Write down the deformation gradient in the specimen in terms of   L / L0
3.10.1.2. Let   e p denote the decomposition of stretch in to elastic and plastic parts. Write
down the elastic and plastic parts of the deformation gradient in terms of e ,  p and find
expressions for the elastic and plastic parts of the stretch rate in terms of e ,  p
3.10.1.3.
3.10.1.4.
3.10.1.5.
3.10.1.6.
Assume that the material can be idealized using Arruda-Boyce potentials


1
11
1
 K
2
U     ( I1  3) 
( I12  9) 
( I13  27)  ...   J  1
2
4
20
1050


2
 2


1
11
1

UT  T  ( I1  3) 
( I12  9) 
( I13  27)  ...
2
4
2
20T
1050T




Obtain an expression for the stress in the specimen in terms of e ,  p , using only the first
two term in the expansion for simplicity. Your answer should include an indeterminate
hydrostatic part.
Calculate the deviatoric stress measure
 1  U
U 
I e UT
1 UT e e 
 ij  2  2 / 3  Te  I1e Te  Bije  1


Bik Bkj 
ij
3 I1e
 J e  I1

I 2 
J e4 / 3 I 2e
in terms of e , and hence find an expression for  p in terms of e
Suppose that the specimen is subjected to a harmonic cycle of nominal strain such that
L   L0 sin t . Use the results of 3.10.1.2 and 3.10.1.4 to obtain a nonlinear differential
equation for e
Use the material data given in Section 3.10.5 to calculate (numerically) the variation of
Cauchy stress in the solid with time induced by cyclic straining. Plot the results as a curve
of Cauchy stress as a function of true strain. Obtain results for various values of  and
frequency  .
35
3.11. Critical State Models for Soils
3.11.1. A drained specimen of a soil can be idealized as Cam-clay, using the constitutive equations listed in
Section 3.11. At time t=0 the soil has a strength a0 . The specimen is subjected to a monotonically
increasing hydrostatic stress p, and the volumetric strain V / V   kk is measured. Calculate a
relationship between the pressure and volumetric strain, in terms of the initial strength of the soil a0
and the hardening rate c.
3.11.2. An undrained specimen of a soil can be idealized as Cam-clay, using the constitutive equations
listed in Section 3.11. The elastic constants of the soil are characterized by its bulk modulus K and
Poisson’s ratio  , while its plastic properties are characterized by M and c. The fluid has a bulk
modulus K w . At time t  0 the soil has a cavity volume fraction n0 and strength a0 , and
ps  pw  0 . The specimen is subjected to a monotonically increasing hydrostatic pressure p, and is
then unloaded. The volumetric strain V / V   kk is measured. Assume that both elastic and plastic
strains are small. Show that the relationship between the normalized pressure p / a0 and normalized
volumetric strain K  kk / a0 is a function of only three dimensionless material properties:
  K w / n0 a0c ,   K / a0 c and   n0 c . Plot the dimensionless pressure-volume curves (showing
both the elastic and plastic parts of the loading cycle for a
Critical
few representative values of  ,  and  .
State
q Dry
3
3.11.3. A drained specimen of Cam-clay is first subjected to a
2
Wet
monotonically increasing confining pressure p, with
1
p

a
maximum value
The confining pressure is then
0.
Ma0
a0
2da
held constant, and the specimen is subjected to a
p
monotonically increasing shear stress q.
Calculate the
p
p
volumetric strain  kk
and the shear strain 12
during the
shear loading as a function of q and appropriate material
properties, and plot the resulting shear stress-shear strain
and volumetric strain-shear strain curves as indicated in the
figure.
Shear stress q
2
3
1
Shear strain 
Volumetric
strain -v
36
3.12. Crystal Plasticity
3.12.1. Draw an inverse pole figure for an fcc single crystal, with [100], [010] and [001] directions parallel
to the {i,j,k} directions, showing the following:
3.12.1.1. The trace of the (010), (100), (110) and (110) planes
3.12.1.2. The trace of the {111} planes
3.12.1.3. The twelve slip directions, labeled
[010]
according to the convention given in
Section 3.12.2 (i.e. a1 , a2 , a3 , etc)
3.12.2. Consider the inverse pole figure for an fcc
crystal with [100], [010] and [001] directions
parallel to the {i,j,k} directions, as shown in the
figure. Show that the circle corresponding to the
trace of the (101) plane has radius R  2 , and is
R
[001]
[100]
centered at the point corresponding to the [101] as
shown in the figure. The simplest approach to this
problem is to note that the [010] direction and the
[101] direction both lie on the circle.
Trace of (101)
[100]
[101]
[101]
[010]
Trace of (101)
3.12.3. Plot a contour map on the standard triangle of the inverse pole figure for an fcc single crystal,
showing the magnitude of the resolved shear stress induced by uniaxial tensile stress on the critical (d1)
slip system. Find the orientation of the tensile axis that gives the largest resolved shear stress
3.12.4. An fcc single crystal deforms by shearing at rate  on the d1 (111) [011] and  on the c2 (111)
[10 1] slip systems.
3.12.4.1. Show that the material fiber parallel to the [112] direction has zero angular velocity;
3.12.4.2. Calculate the rate of stretching of the material fiber parallel to the [112] direction.
3.12.5. A single crystal is loaded in uniaxial tension. The direction of
the loading axis, specified by a unit vector p remains fixed during m
straining. The crystal deforms by slip on a single system, with slip
direction s and slip plane normal m . The deformation gradient
resulting from a shear strain 

is


Fij  Rik  kj    s
k mj
-1
m* tan 
p
p
s*
s

where Rij is a proper orthogonal tensor (i.e. det(R)=1, Rik R jk   ij ), representing a rigid rotation.
Assume that that the material fiber parallel to the loading axis does not rotate during deformation.
Show that:
Rij  ij cos  (1  cos )ni n j  sin  ikj nk
where
ni ijk s j pk
cos  (1    pi si pk m
k )/C ,
C  1    2 ( pi mi )2  2  pi si pk m
k .


sin     ( pi mi ) 1  ( pi si ) 2 / C ,
and
37
3.12.6. Consider the single crystal loaded in uniaxial tension described in the preceding problem. Calculate
3.12.6.1. The angular velocity vector that describes the angular velocity of the slip direction and slip
plane normal (it is easiest to do this by first assuming that the slip direction and slip plane
normal are fixed, and calculating the angular velocity of the material fiber parallel to p, and
hence deducing the angular velocity that must be imposed on the crystal to keep p pointing
in the same direction)
3.12.6.2. Calculate the spin tensor W associated with the angular velocity calculated in 3.10.6.1
3.12.7. The resolved shear stress on a slip system in a crystal is related to the Kirchhoff stress τ by
   s*  τ  m* . Show that the rate of change of resolved shear stress is related to the elastic
Jaumann rate of Kirchhoff stress by
 e

d 
e
 si*  ij  Dik
 kj  Dike  kj  m*j
dt


3.12.8. A rigid –perfectly plastic single crystal contains two slip
systems, oriented at angles 1 and 2 as illustrated in the
figure. The solid is deformed in simple shear as indicated
3.12.8.1. Suppose that 1  600 , 2  1200 . Sketch the
yield locus (in 11 ,  22 space) for the crystal.
3.12.8.2. Write down the velocity gradient L in the strip,
and compute an expression the deformation rate
D. Hence, show that (at the instant shown) the
slip rates on the two slip systems are given by
v0
s(2)
(2)
m
m(1)
s(1)


x2
h
x1
 (1)   sin 22 / sin 2(2  1 )
 (2)   sin 21 / sin 2(1  2 )
and give an expression for  in terms of v0 and h.
3.12.8.3.
Assume that 11   22  0 , and that the slip systems have critical resolved shear stress  c .
Show that the stress in the crystal is
11   22   c (sin 22  sin 21 ) / sin 2(2  1)
12   c (cos 22  cos 21 ) / sin 2(2  1)
3.12.9. Consider a single crystal of copper, with constitutive equation given in section 3.12.5. and
properties listed in 3.12.6.
3.12.9.1. Plot a graph showing the uniaxial true stress-v-true strain curve for a crystal that is loaded
parallel to the [112] direction
3.12.9.2. Plot a graph showing the uniaxial true stress-v-true strain curve for a crystal loaded parallel
to the [111] direction
3.12.9.3. Plot a graph showing the uniaxial true stress-v-true strain curve for a crystal loaded parallel
to the [001] direction.
38
3.12.10.A rate independent, rigid perfectly plastic fcc single crystal is loaded in uniaxial tension, with
tensile axis parallel to the [102] crystallographic direction. Assume that the crystal rotates to maintain
the material fiber parallel to the [102] direction aligned with the tensile axis.
3.12.10.1. Assuming the magnitude of the shearing rate is  on all active slip systems, calculate the
velocity gradient in the crystal, expressing your answer as components in a basis aligned
with the {100} directions.
3.12.10.2. Hence, show that the [102] loading direction is not a stable orientation – i.e. the tensile axis
rotates with respect to the crystallographic directions.
3.12.10.3. Calculate the instantaneous stretching rate of the tensile axis as a function of the magnitude
of the shearing rate.
3.12.10.4. Deduce the angular velocity vector that characterizes the instantaneous rotation of the
crystal relative to the tensile axis.
3.12.10.5. Show the motion of the tensile axis on an inverse pole figure. Without calculations, predict
the eventual steady-state orientation of the tensile axis with respect to the loading axis.
3.13. Constitutive Laws for Contacting Surfaces and Interfaces in Solids
3.13.1. The figure shows two elastic blocks with Young’s modulus E that are bonded
together at an interface. The interface can be characterized using the reversible
constitutive law described in Section 3.13.1. The top block is subjected to
tractions which induce a uniaxial stress  22   in the blocks, a separation  n at
the interface, and a displacement U at the surface of the upper block.
3.13.1.1. Show that the stress and displacement can be expressed in
dimensionless form as
  


U

 n exp 1  n 
1

 max  n
n
 max
 n 
3.13.1.2.
3.13.1.3.

U
a
n
a
where   a max /  n E .
Plot graphs of  /  max and  n /  n as functions of U /  n for various values of  (it is
easiest to do a parametric plot). Hence, show that if  is less than a critical value, the
interface separates smoothly under monotonically increasing U /  n . In contrast, if 
exceeds the critical value, the interface suddenly snaps apart (with a sudden drop in stress)
at a critical value of U /  n . (Under decreasing U /  n the interface re-adheres, with a
similar transition from smooth attachment to sudden snapping at a critical  ). Give an
expression for the critical value of  .
Plot a graph showing the critical displacement U /  n at separation and attachment as a
function of  .
3.13.2. Two rigid surfaces slide against one another under an applied shear force T and a normal force N.
The interface may be characterized using the rate and state dependent friction law described in Section
3.13.2. The blocks slide at speed V1 until the friction force reaches its steady state value. The sliding
speed is then increased instantaneously to a new value V2 . Calculate an expression for the variation of
the friction force T as a function of the distance slid d.
39
3.13.3. A rigid block with length L slides over a flat rigid surface at
constant speed V under an applied shear force T and a normal force N.
The contacting surfaces may be idealized using the rate and state
dependent friction law described in Section 3.13.2. The state variables
p    0 for any point on the stationary surface that lies ahead of the
rigid block. Calculate the shear force T as a function of the length of
the block, the sliding speed, and relevant material properties.
N
T
V
L
3.13.4. A rigid block with mass M is pulled over a flat surface by a
k
V
spring with stiffness k. The end of the spring is pulled at a
M
steady speed V. The contacting surface may be idealized using
the rate and state dependent friction law described in Section
x
3.13.2. Take tv   and ke   in the constitutive equation,
for simplicity. Obtain a governing equation for the rate of change of length of the spring dx / dt in
terms of V, M, and the properties of the interface, for the limiting case. Hence, investigate the stability
of steady sliding.
40
Chapter 4
Solutions to simple boundary and initial value problems
4.
4.1. Axially and Spherically Symmetric Solutions for Linear Elastic Solids
4.1.1. A solid cylindrical bar with radius a and length L is subjected to a
uniform pressure p on its ends. The bar is made from a linear elastic solid
with Young’s modulus E and Poisson’s ratio  .
4.1.1.1.
Write down the components of the stress in the bar. Show that
the stress satisfies the equation of static equilibrium, and the
boundary conditions  ij ni  t j on all its surfaces. Express
4.1.1.2.
4.1.1.3.
4.1.1.4.
4.1.1.5.
4.1.1.6.
4.1.1.7.
your answer as components in a Cartesian basis {e1, e2 , e3}
with e1 parallel to the axis of the cylinder.
Find the strain in the bar (neglect temperature changes)
Find the displacement field in the bar
Calculate a formula for the change in length of the bar
Find a formula for the stiffness of the bar (stiffness = force/extension)
Find the change in volume of the bar
Calculate the total strain energy in the bar.
4.1.2. Elementary calculations predict that the stresses in a internally
pressurized thin-walled sphere with radius R and wall thickness
t<<R are     pR / 2t  rr  p / 2 . Compare this estimate
with the exact solution in Section 4.1.4. To do this, set
a  R[1  t /(2R)] b  R[1  t /(2R)] and expand the formulas for the
stresses as a Taylor series in t/R. Suggest an appropriate range of
t/R for the thin-walled approximation to be accurate.
4.1.3. A baseball can be idealized as a small rubber core with radius a,
surrounded by a shell of yarn with outer radius b. As a first
approximation, assume that the yarn can be idealized as a linear elastic
solid with Young’s modulus Es and Poisson’s ration  s , while the
core can be idealized as an incompressible material. Suppose that ball
is subjected to a uniform pressure p on its outer surface. Note that, if
the core is incompressible, its outer radius cannot change, and
therefore the radial displacement u R  0 at R  a . Calculate the full
displacement and stress fields in the yarn in terms of p and relevant
geometric variables and material properties.
41
4.1.4. Reconsider problem 3, but this time assume that the core is to be idealized as a linear elastic solid
with Young’s modulus Ec and Poisson’s ration  c . Give expressions for the displacement and stress
fields in both the core and the outer shell.
4.1.5. Suppose that an elastic sphere, with outer radius a   , and with Young’s modulus E and
Poisson’s ratio  is inserted into a spherical shell with identical elastic properties, but with inner radius
a and outer radius b . Assume that   a so that the deformation can be analyzed using linear
elasticity theory. Calculate the stress and displacement fields in both the core and the outer shell.
4.1.6. A spherical planet with outer radius a has a radial variation in its
density that can be described as
 ( R)  0 a 2 /(a 2  R 2 )
As a result, the interior of the solid is subjected to a radial body force field
ga
a

1 R 
b
1  tan
 er
R(1   / 4)  R
a
where g is the acceleration due to gravity at the surface of the sphere.
Assume that the planet can be idealized as a linear elastic solid with
Young’s modulus E and Poisson’s ratio  . Calculate the displacement and stress fields in the solid.
4.1.7. A solid, spherical nuclear fuel pellet with outer radius a is subjected to a uniform internal
distribution of heat due to a nuclear reaction. The heating induces a steady-state temperature field
r2
T (r )  Ta  T0 
 T0
a2
where T0 and Ta are the temperatures at the center and outer surface of the pellet, respectively.
Assume that the pellet can be idealized as a linear elastic solid with Young’s modulus E , Poisson’s
ratio  and thermal expansion coefficient  . Calculate the distribution of stress in the pellet.
4.1.8. A long cylindrical pipe with inner radius a and outer radius b has hot
fluid with temperature Ta flowing through it. The outer surface of the
pipe has temperature Tb . The inner and outer surfaces of the pipe are
traction free. Assume plane strain deformation, with  zz  0 . In addition,
assume that the temperature distribution in the pipe is given by
T log(r / b)  Tb log(r / a)
T (r )  a
log(a / b)
4.1.8.1.
Calculate the stress components  rr ,  , zz in the pipe.
4.1.8.2.
Find a formula for the variation of Von-Mises stress
1
e 
1   2 2  1   3 2   2   3 2
2
in the tube. Where does the maximum value occur?
4.1.8.3.
The tube will yield if the von Mises stress reaches the yield stress of the material. Calculate
the critical temperature difference Ta  Tb that will cause yield in a mild steel pipe.


42
4.2. Axially and spherically symmetric solutions to quasi-static elastic-plastic
problems
4.2.1. The figure shows a long hollow cylindrical shaft with inner radius a and
outer radius b, which spins with angular speed  about its axis. Assume that
the disk is made from an elastic-perfectly plastic material with yield stress Y
and density  . The goal of this problem is to calculate the critical angular
speed that will cause the cylinder to collapse (the point of plastic collapse
occurs when the entire cylinder reaches yield).
4.2.1.1.
Using the cylindrical-polar basis shown, list any stress or strain
components that must be zero. Assume plane strain deformation.
4.2.1.2.
Write down the boundary conditions that the stress field must
satisfy at r=a and r=b
4.2.1.3.
Write down the linear momentum balance equation in terms of the
stress components, the angular velocity and the disk’s density. Use polar coordinates and
assume axial symmetry.
4.2.1.4.
Using the plastic flow rule, show that  zz  ( rr   ) / 2 if the cylinder deforms
plastically under plane strain conditions
4.2.1.5.
Using Von-Mises yield criterion, show that the radial and hoop stress must satisfy
   rr  2Y / 3
4.2.1.6.
Hence, show that the radial stress must satisfy the equation
d rr
2 Y
   r 2 
dr
3 r
4.2.1.7.
Finally, calculate the critical angular speed that will cause plastic collapse.
4.2.2. Consider a spherical pressure vessel subjected to cyclic internal pressure, as described in Section
4.2.4. Show that a cyclic plastic zone can only develop in the vessel if b / a exceeds a critical
magnitude. Give a formula for the critical value of b / a , and find a (numerical, if necessary) solution
for b / a .
4.2.3. A long cylindrical pipe with internal bore a and outer diameter b is made from an elastic-perfectly
plastic solid, with Young’s modulus E, Poisson’s ratio  and uniaxial tensile yield stress Y is subjected
to internal pressure. The (approximate) solution for a cylinder that is subjected to a monotonically
increasing pressure is given in Section 4.2.6. The goal of this problem is to extend the solution to
investigate the behavior of a cylinder that is subjected to cyclic pressure.
4.2.3.1.
Suppose that the internal pressure is first increased to a value that lies in the range
(1  a 2 / b 2 )  3 pa / Y  2log(b / a) , and then returned to zero. Assume that the solid
unloads elastically (so the change in stress during unloading can be calculated using
4.2.3.2.
4.2.3.3.
the elastic solution). Calculate the residual stress in the cylinder after unloading.
Hence, determine the critical internal pressure at which the residual stresses cause
the cylinder to yield after unloading
Find the stress and displacement in the cylinder at the instant maximum pressure, and after
subsequent unloading, for internal pressures exceeding the value calculated in 4.2.3.2.
43
4.2.4. The following technique is sometimes used to
connect tubular components down oil wells. As
manufactured, the smaller of the two tubes has inner and
outer radii (a, b) , while the larger has inner and outer
radii (b, d ) , so that the end of the smaller tube can
simply be inserted into the larger tube. An over-sized die
is then pulled through the bore of the inner of the two
tubes. The radius of the die is chosen so that both
cylinders are fully plastically deformed as the die passes
through the region where the two cylinders overlap. As
a result, a state of residual stress is developed at the
coupling, which clamps the two tubes together. Assume
that the tubes are elastic-perfectly plastic solids with
Young’s modulus E, Poisson’s ratio  and yield stress
in uniaxial tension Y.
4.2.4.1.
Use the solution given in Section 4.2.6 to
calculate the radius of the die that will cause
both cylinders to yield throughout their wall-thickness (i.e. the radius of the plastic zone
must reach d).
4.2.4.2.
The die effectively subjects to the inner bore of the smaller tube to a cycle of pressure. Use
the solution to the preceding problem to calculate the residual stress distribution in the
region where the two tubes overlap (neglect end effects and assume plane strain
deformation)
4.2.4.3.
For d / a  1.5 , calculate the value of b that gives the strongest coupling.
4.2.5. A spherical pressure vessel is subjected to internal pressure
pa and is free of traction on its outer surface. The vessel deforms
by creep, and may be idealized as an elastic-power law
viscoplastic solid with flow potential
m
 
g ( e )   0  e 
 Y 
where Y , m,  0 are material properties and  e is the Von-Mises
equivalent stress. Calculate the steady-state stress and strain rate
fields in the solid, and deduce an formula for the rate of expansion
of the inner bore of the vessel. Note that at steady state, the stress is constant, and so the elastic strain
rate must vanish.
44
4.3. Spherically and axially symmetric solutions to quasi-static large strain
elasticity problems
4.3.1. Consider the pressurized hyperelastic spherical shell described in Section 4.3.3. For simplicity,
assume that the shell is made from an incompressible neo-Hookean material (recall that the NeoHookean constitutive equation is the special case  2  0 in the Mooney-Rivlin material). Calculate the
total strain energy of the sphere, in terms of relevant geometric and material parameters. Hence, derive
an expression for the total potential energy of the system (assume that the interior and exterior are
subjected to constant pressure). Show that the relationship between the internal pressure and the
geometrical parameters   a / A ,   b / B can be obtained by minimizing the potential energy of the
system.
4.3.2. Consider an internally pressurized hollow rubber cylinder, as shown in
the picture. Assume that
Before deformation, the cylinder has inner radius A and outer radius B
After deformation, the cylinder has inner radius a and outer radius b
The solid is made from an incompressible Mooney-Rivlin solid, with
strain energy potential


U  1 ( I1  3)  2 ( I 2  3)
2
2
No body forces act on the cylinder; the inner surface r=a is subjected to
pressure pa ; while the outer surface r=b is free of stress.
Assume plane strain deformation.
Assume that a material particle that has radial position R before deformation moves to a position r=f(R)
after the cylinder is loaded. This problem should be solved using cylindrical-polar coordinates.
4.3.2.1.
Find an expression for the deformation gradient F in terms of f(R) and R
4.3.2.2.
Express the incompressibility condition det(F)=1 in terms of f(R)
4.3.2.3.
Integrate the incompressibility condition to calculate r in terms of R¸A and a, and also
calculate the inverse expression that relates R to A, a and r.
4.3.2.4.
Calculate the components of the left Cauchy-Green deformation tensor Brr , B
4.3.2.5.
Find an expression for the Cauchy stress components  rr , in the cylinder in terms of
Brr , B , Bzz and an indeterminate hydrostatic stress p.
4.3.2.6.
Use 3.4 and the equilibrium equation to derive an expression for the radial stress  rr in the
cylinder. Use the boundary conditions to find a relationship between the applied pressure
and   a / A ,   b / B .
4.3.2.7.
Plot a graph showing the variation of normalized pressure pa B 2 /( 1  2 )( B 2  A2 ) as a
function of the normalized displacement of the inner bore of the cylinder a / A  1 .
Compare the nonlinear elastic solution with the equivalent linear elastic solution.
45
4.3.3. A long rubber tube has internal radius A and external radius B. The tube can be idealized as an
incompressible neo-Hookean material with material constant 1 . The tube is turned inside-out, so that
the surface that lies at R=A in the undeformed configuration moves to r=a in the deformed solid, while
the surface that lies at R=B moves to r=b. Note that B>A, and a>b. To approximate the deformation,
assume that
planes that lie perpendicular to e z in the undeformed solid remain perpendicular to e z after
deformation
The axial stretch  zz in the tube is constant
It is straightforward to show that the deformation mapping can be described as
r  a 2  ( B 2  R 2 ) / zz
4.3.3.1.
4.3.3.2.
4.3.3.3.
4.3.3.4.
4.3.3.5.
R  B 2  (a 2  r 2 )zz
z  zz Z
Z   z / zz
Calculate the deformation gradient, expressing your answer in terms of R, as components in
the {e r , e , e z } basis. Verify that the deformation preserves volume.
Calculate the components of the left Cauchy-Green deformation tensor Brr , B , Bzz
Find an expression for the Cauchy stress components  rr ,  , zz in the cylinder in terms
of Brr , B and an indeterminate hydrostatic stress p.
Use 4.3 and the equilibrium equation and boundary conditions to calculate an expression
for the Cauchy stress components.
Finally, use the condition that the resultant force acting on any cross-section of the tube
must vanish to obtain an equation for the axial stretch  zz . Does the tube get longer or
shorter when it is inverted?
4.3.4. Two spherical, hyperelastic shells are connected
by a thin tube, as shown in the picture. When stress
free, both spheres have internal radius A and external
radius B. The material in each sphere can be
idealized as an incompressible, neo-Hookean solid,
with material constant 1 . Suppose that the two
spheres together contain a volume V  8 A3 / 3 of an
incompressible fluid. As a result, the two spheres
have deformed internal and external radii (a1 , b1 ) , (a2 , b2 ) as shown in the picture. Investigate the
possible equilibrium configurations for the system, as functions of the dimensionless fluid volume
  V /(8 A3 / 3)  1 and B/A. To display your results, plot a graph showing the equilibrium values of
1  a1 / A as a function of  , for various values of B/A. You should find that for small values of 
there is only a single stable equilibrium configuration. For  exceeding a critical value, there are three
possible equilibrium configurations: two in which one sphere is larger than the other (these are stable),
and a third in which the two spheres have the same size (this is unstable).
46
4.3.5. In a model experiment intended to duplicate the propulsion
mechanism of the lysteria bacterium, a spherical bead with radius a
is coated with an enzyme known as an “Arp2/3 activator.” When
suspended in a solution of actin, the enzyme causes the actin to
polymerize at the surface of the bead. The polymerization reaction
causes a spherical gel of a dense actin network to form around the
bead. New gel is continuously formed at the bead/gel interface,
forcing the rest of the gel to expand radially around the bead. The
actin gel is a long-chain polymer and consequently can be
idealized as a rubber-like incompressible neo-Hookean material.
Experiments show that after reaching a critical radius the actin gel
loses spherical symmetry and occasionally will fracture. Stresses in the actin network are believed to
drive both processes. In this problem you will calculate the stress state in the growing, spherical, actin
gel.
4.3.5.1.
Note that this is an unusual boundary value problem in solid mechanics, because a
compatible reference configuration cannot be identified for the solid. Nevertheless, it is
possible to write down a deformation gradient field that characterizes the change in shape
of infinitesimal volume elements in the gel.
To this end: (a) write down the length of a
circumferential line at the surface of the bead; (b) write down the length of a
circumferential line at radius r in the gel; (c) use these results, together with the
incompressibility condition, to write down the deformation gradient characterizing the
shape change of a material element that has been displaced from r=a to a general position
r. Assume that the bead is rigid, and that the deformation is spherically symmetric.
4.3.5.2.
Suppose that new actin polymer is generated at volumetric rate V .
Use the
incompressibility condition to write down the velocity field in the actin gel in terms of V , a
and r (think about the volume of material crossing a radial line per unit time)
4.3.5.3.
Calculate the velocity gradient v   in the gel (a) by direct differentiation of 5.2 and (b)
by using the results of 5.1. Show that the results are consistent.
4.3.5.4.
Calculate the components of the left Cauchy-Green deformation tensor field and hence
write down an expression for the Cauchy stress field in the solid, in terms of an
indeterminate hydrostatic pressure.
4.3.5.5.
Use the equilibrium equations and boundary condition to calculate the full Cauchy stress
distribution in the bead. Assume that the outer surface of the gel (at r=b) is traction free.
4.3.6. A rubber sheet is wrapped around a rigid cylindrical shaft with
radius a. The sheet has thickness t, and can be idealized as an
incompressible neo-Hookean solid. A constant tension T per unit
out-of-plane distance is applied to the sheet during the wrapping
process. Calculate the full stress field in the solid rubber, and find an
expression for the radial pressure acting on the shaft. Assume that
t / b  0 and neglect the shear stress component  r .
T
t
b
a
47
4.4. Solutions to simple dynamic problems involving linear elastic solids
4.4.1. Calculate longitudinal and shear wave speeds in (a) Aluminum nitride; (b) Steel; (d) Aluminum and
(e) Rubber.
4.4.2. A linear elastic half-space with Young’s modulus E and Poisson’s ratio  is stress free and
stationary at time t=0¸ is then subjected to a constant pressure p0 on its surface for t>0.
4.4.2.1.
Calculate the stress, displacement and velocity in the solid as a function of time
4.4.2.2.
Calculate the total kinetic energy of the half-space as a function of time
4.4.2.3.
Calculate the total potential energy of the half-space as a function of time
4.4.2.4.
Verify that the sum of the potential and kinetic energy is equal to the work done by the
tractions acting on the surface of the half-space.
4.4.3. The surface of an infinite linear elastic half-space with Young’s modulus E and Poisson’s ratio  is
subjected to a harmonic pressure on its surface, given by p(t )  p0 sin t t>0, with p=0 for t<0.
4.4.3.1.
Calculate the distribution of stress, velocity and displacement in the solid.
4.4.3.2.
What is the phase difference between the displacement and pressure at the surface?
4.4.3.3.
Calculate the total work done by the applied pressure in one cycle of loading.
4.4.4. A linear elastic solid with Young’s modulus E Poisson’s ratio  and density  is bonded to a rigid
solid at x1  a . Suppose that a plane wave with displacement and stress field
 (1  2 )(1   )  0
(cLt  x1 )

(1  )
E
u2 ( x1, t )  
0

x1  cLt
x1  cLt
 0
0
11  
x1  cLt
x1  cLt
is induced in the solid, and at time t  x1 / a is reflected off the interface. Find the reflected wave, and
sketch the variation of stress and velocity in the elastic solid just before and just after the reflection
occurs.
4.4.5. Consider the plate impact experiment described in Section 4.4.8
4.4.5.1.
Draw graphs showing the stress and velocity at the impact face of the flyer plate as a
function of time.
4.4.5.2.
Draw graphs showing the stress and velocity at the rear face of the flyer plate as a function
of time
4.4.5.3.
Draw graphs showing the stress and velocity at the mid-plane of the flyer plate as a
function of time
4.4.5.4.
Draw a graph showing the total strain energy and kinetic energy of the system as a function
of time. Verify that total energy is conserved.
4.4.5.5.
Draw a graph showing the total momentum of the flyer plate and the target plate as a
function of time. Verify that momentum is conserved.
v0
4.4.6. In a plate impact experiment, two identical elastic plates with thickness h, Young’s
modulus E, Poisson’s ratio  , density  and longitudinal wave speed c L are caused to
collide, as shown in the picture. Just prior to impact, the projectile has a uniform velocity
v0 . Draw the (x,t) diagram for the two solids after impact. Show that the collision is
perfectly elastic, in the terminology of rigid body collisions, in the sense that all the energy
in the flyer is transferred to the target.
Target
h
h
e2
e1
Flyer
48
v0
4.4.7. In a plate impact experiment, an elastic plates with thickness h, Young’s modulus
E, Poisson’s ratio  , density  and longitudinal wave speed c L impacts a second
plate with identical elastic properties, but thickness 2h, as shown in the picture. Just
prior to impact, the projectile has a uniform velocity v0 . Draw the (x,t) diagram for
the two solids after impact.
Target
h
2h
e2
e1
Flyer
4.4.8. A “ Split-Hopkinson bar” or
“Kolsky bar” is an apparatus that
Stress wave
a
is used to measure plastic flow in
d/2
d/2
d/2
materials at high rates of strain (of p
Strain gage
Strain gage
order 1000/s). The apparatus is
sketched in the figure. A small
Incident bar
Specimen
Transmitted bar
specimen of the material of
interest, with length a<<d, is
placed between two long slender bars with length d. Strain gages
d/2cB d/c
B
are attached near the mid-point of each bar. At time t=0 the
time
system is stress free and at rest. Then, for t>0 a constant
pressure p is applied to the end of the incident bar, sending a
plane wave down the bar. This wave eventually reaches the
I
R
specimen. At this point part of the wave is reflected back up the
Incident bar
d/cB
time
incident bar, and part of it travels through the specimen and into
the second bar (known as the ‘transmission bar’). The history of
T
stress and strain in the specimen can be deduced from the history
of strain measured by the two strain gages. For example, if the
Transmitted bar
specimen behaves as an elastic-perfectly plastic solid, the
incindent and reflected gages would record the data shown in the
figure. The goal of this problem is to calculate a relationship between the measured strains and the
stress and strain rate in the specimen. Assume that the bars are linear elastic with Young’s modulus E
and density  , and wave speed cB  E /  , and that the bars deform in uniaxial compression.
4.4.8.1.
Write down the stress, strain and velocity field in the incident bar as a function of time and
distance down the bar in terms of the applied pressure p and relevant material and
geometric parameters, for t  d / cB .
4.4.8.2.
Assume that the waves reflected from, and transmitted through, the specimen are both plane
waves. Let  R and  T denote the compressive strains in the regions behind the reflected
and transmitted wave fronts, respectively. Write down expressions for the stress and
velocity behind the wave fronts in both incident and transmitted bars in terms of  R and
 T , for 2d / cB  t  d / cB
4.4.8.3.
The stress behind the reflected and transmitted waves must equal the stress in the specimen.
In addition, the strain rate in the specimen can be calculated from the relative velocity of
the incident and transmitted bars where they touch the specimen. Show that the strain rate
in the specimen can be calculated from the measured strains as ( I   R   T )cL / a , while
the stress in the specimen can be calculated from   E T .
49
v0
4.4.9. In a plate impact experiment, two plates with identical thickness h, Young’s modulus
E, Poisson’s ratio  are caused to collide, as shown in the picture. The target plate has
twice the mass density of the flyer plate. Find the stress and velocity behind the waves
generated by the impact in both target and flyer plate. Hence, draw the (x,t) diagram for
the two solids after impact.
Target
h
h


e2
e1
Flyer
v0
4.4.10. In a plate impact experiment, two plates with identical thickness h, Young’s
modulus E, Poisson’s ratio  , and density  are caused to collide, as shown in the
picture. The flyer plate has twice the mass density of the target plate. Find the stress and
velocity behind the waves generated by the impact in both target and flyer plate. Hence,
draw the (x,t) diagram for the two solids after impact.
Target
h
h
 
e2
e1
Flyer
4.4.11. The figure shows a pressure-shear plate impact experiment. A flyer
plate with speed v0 impacts a stationary target. Both solids have identical
thickness h, Young’s modulus E, Poisson’s ratio  , density  and
longitudinal and shear wave speeds c L and cS . The faces of the plates
are inclined at an angle  to the initial velocity, as shown in the figure.
Both pressure and shear waves are generated by the impact. Let {e1 , e 2 }
denote unit vectors Let 11   0 denote the (uniform) stress behind the
propagating pressure wave in both solids just after impact, and 12   0
denote the shear stress behind the shear wave-front. Similarly, let
v0
Target
e2

e1
h
h
Flyer
v1f , v2f denote the change in longitudinal and transverse velocity in the flier across the pressure and
shear wave fronts, and let v1t , v2t denote the corresponding velocity changes in the target plate.
Assume that the interface does not slip after impact, so that both velocity and stress must be equal in
both flier and target plate at the interface just after impact. Find expressions for  0 ,  0 , v1f , v2f ,
v1t , v2t in terms of v0 , and relevant material properties.
4.4.12. Draw the full (x,t) diagram for the pressure-shear configuration described in problem 11. Assume
that the interface remains perfectly bonded until it separates under the application of a tensile stress.
Note that you will have to show (x,t) diagrams associated with both shear and pressure waves.
50
4.4.13. Consider an isotropic, linear elastic solid with Young’s
e2
modulus E, Poisson’s ratio  , density  and shear wave speed
e1
c S . Suppose that a plane, constant stress shear wave propagates
through the solid, which is initially at rest. The wave propagates


in a direction p  cos e1  sin  e2 , and the material has particle
C B A
velocity v  Ve3 behind the wave-front.
4.4.13.1. Calculate the components of stress in the solid
Reflected
Incident
behind the wave front.
wave front
wave front
4.4.13.2. Suppose that the wave front is incident on a flat,
stress free surface. Take the origin for the coodinate system at some arbitrary time t at the
point where the propagating wave front just intersects the surface, as shown in the picture.
Write down the velocity of this intersection point (relative to a stationary observer) in terms
of V and  .
4.4.13.3. The surface must be free of traction both ahead and behind the wave front. Show that the
boundary condition can be satisfied by superposing a second constant stress wave front,
which intersects the free surface at the origin of the coodinate system defined in 13.2, and
propagates in a direction p  cos e1  sin  e2 . Hence, write down the stress and particle
velocity in each of the three sectors A, B, C shown in the figure. Draw the displacement of
the free surface of the half-space.
e
2
4.4.14. Suppose that a plane, constant stress pressure wave
propagates through an isotropic, linear elastic solid that is
e1
initially at rest. The wave propagates in a direction

p  cos e1  sin  e2 , and the material has particle velocity
D 

A
v  Vp behind the wave-front.
C
B
4.4.14.1. Calculate the components of stress in the solid Reflected
Incident
P wave front Reflected
behind the wave front.
P wave front
S wave front
4.4.14.2. Suppose that the wave front is incident on a flat,
stress free surface. Take the origin for the
coodinate system at some arbitrary time t at the point where the propagating wave front just
intersects the surface, as shown in the picture. Write down the velocity of this intersection
point (relative to a stationary observer) in terms of V and  .
4.4.14.3. The pressure wave is reflected as two waves – a reflected pressure wave, which propagates
in direction p  cos e1  sin  e2 and has particle velocity v p p and a reflected shear wave,
which

4.4.14.4.
4.4.14.5.
propagates in direction

p  cos1e1  sin 1e2
and
has
particle
velocity
vs sin 1 e1  cos1e2 . Use the condition that the incident wave and the two reflected
waves must always intersect at the same point on the surface to write down an equation for
1 in terms of  and Poisson’s ratio.
The surface must be free of traction. Find equations for v p and vs in terms of V,  , 1 and
Poisson’s ratio.
Find the special angles for which the incident wave is reflected only as a shear wave (this is
called “mode conversion”)
51
Chapter 5
Analytical Techniques and Solutions for Linear Elastic Solids
5.
5.1. General Principles
5.1.1. A spherical shell is simultaneously subjected to internal
pressure, and is heated internally to raise its temperature at r  a
to a temperature Ta , while at r  b its surface is traction free, and
temperature is Tb . Use the principle of superposition, together
with the solutions given in Chapter 4.1, to determine the stress
field in the sphere.
5.1.2. The stress field around a cylindrical hole in an infinite solid,
e2
which is subjected to uniaxial tension 11   0 far from the hole, is
given by
r
  3a 4 a 2 

3a 2

11   0 1   4  2  cos 4  2 cos 2 
a

  2r

r 
2r
 

e1
2
4
2
 a

3a 
a
 22   0   2  4  cos 4  2 cos 2 
 r

2r 
2r


  3a 4 a 2 

a2
12   0   4  2  sin 4  2 sin 2 
  2r

r 
2r


Using the principle of superposition, calculate the stresses near a hole in a solid which is subjected to
shear stress 12   0 at infinity.
52
5.1.3. The stress field due to a concentrated line load, with force per unit outof-plane distance P acting on the surface of a large flat elastic solid are
given by
x13
x1 x22
x12 x2
2P
2P
2P
11  






22
12
2
2
2



x12  x22
x12  x22
x12  x22





P
r


e2
e1
The stress field due to a uniform pressure distribution acting on a strip with
width 2a is
p
 22    2 1   2    sin 21  sin 2 2  
2
p
11    2 1   2    sin 21  sin 2 2  
2
p
12 
 cos 21  cos 22 
2
where 0     and 1  tan 1 x1 /( x2  a )  2  tan 1 x1 /( x2  a)
p
e2
a
a


e1
Show that, for x12  x22  a the stresses due to the uniform pressure become equal to the stresses induced
by the line force (you can do this graphically, or analytically).
5.1.4. The stress field in an infinite solid that contains a spherical cavity with
radius a at the origin, and is subjected to a uniform uniaxial stress  33   0
far from the sphere is given by

 ij
3a3 xi x j 
x32 
3a3
a2 
a2

3  5  4


6

5


5

10




ij
 0 2(7  5 ) R3 
R 2 
2(7  5 ) R5 
R2
R 2 
 i3 j 3 
3

a5  15a x3 ( x j i 3  xi j 3 )  a 2

(7

5

)

5(1

2

)

3

 



 R2

(7  5 ) 
R3
R5 
(7  5 ) R5


Show that the hole only influences the stress field in a region close to the hole.
a3
e2
a
e1
e3
53
5.2. Airy Function Solution to Plane Stress and Strain Static Linear Elastic
Problems
5.2.1.
A rectangular dam is subjected to pressure p( x2 )  w x2 on one
face, where  w is the weight density of water. The dam is made from
concrete, with weight density c (and is therefore subjected to a body
e1
a
force ce2 per unit volume). The goal is to calculate formulas for a
and L to avoid failure.
5.2.1.1.
Write down the boundary conditions on all four sides of
the dam.
5.2.1.2.
Consider the following approximate state of stress in the
dam
 w x23 x1
A
a
e2
L
B
 w x2 x1
10 x12  6a 2   c x2

4a
20a
 w x2  w x2 x1 2
11  

x1  3a 2 

3
2
4a
 22 
12 
3
3 w x22
8a
3

3
(a 2  x12 ) 
w
8a
3
 a4  x14   320aw (a2  x12 )
Show that (i) The stress state satisfies the equilibrium equations (ii) the stress state exactly
satisfies boundary conditions on the sides x1   a , (iii) The stress does not satisfy the
boundary condition on x2  0 exactly.
5.2.1.3.
5.2.1.4.
5.2.1.5.
Show, however, that the resultant force acting on x2  0 is zero, so by Saint Venant’s
principle the stress state will be accurate away from the top of the dam.
The concrete cannot withstand any tension. Assuming that the greatest principal tensile
stress is located at point A ( x1  a, x2  L) , show that the dam width must satisfy
The concrete fails by crushing when the minimum principal stress reaches 1min   c .
Assuming the greatest principal compressive stress is located at point B, ( x1  a, x2  L)
show that the height of the dam cannot exceed
5.2.2. The stress due to a line load magnitude P per unit out-of-plane
length acting tangent to the surface of a homogeneous, isotropic halfspace can be generated from the Airy function
P
   r cos

Calculate the displacement field in the solid, following the procedure
in Section 5.2.6
e2
P

e1
r
54
5.2.3.
The figure shows a simple design for a dam.
5.2.3.1.
5.2.3.2.
5.2.3.3.
5.2.3.4.
5.2.3.5.
5.2.3.6.
e1
Write down an expression for the hydrostatic pressure
n
Water

in the fluid at a depth x 2 below the surface
density 
Hence, write down an expression for the traction vector
acting on face OA of the dam.
Write down an expression for the traction acting on face
OB
e2
Write down the components of the unit vector normal to
face OB in the basis shown
Hence write down the boundary conditions for the stress state in the dam on faces OA and
OB
Consider the candidate Airy function
C1 3 C2 2
C
C
x1  x1 x2  3 x1 x22  4 x23
6
2
2
6

5.2.3.7.
5.2.3.8.
Is this a valid Airy function? Why?
Calculate the stresses generated by the Airy function given in 5.2.2.6
Use 5.2.2.5 and 5.2.2.7 to find values for the coefficients in the Airy function, and hence
show that the stress field in the dam is
 11 
x2

2

 22  3 x1  2 x2
tan 
tan 


 12  2 x1
tan 
5.2.4.
Consider the Airy function
 

 
a4 
log(r )  0 r 2  0  2a 2  r 2 
 cos 2
2
4
4 
r 2 
0
Verify that the Airy function satisfies the appropriate governing equation. Show that this stress state
represents the solution to a large plate containing a circular hole with radius a at the origin, which is loaded
by a tensile stress  0 acting parallel to the e1 direction. To do this,
5.2.4.1.
5.2.4.2.
5.2.4.3.
Show that the surface of the hole is traction free – i.e.  rr   r  0 on r=a
Show that the stress at r / a   is  rr   0 (1  cos 2 ) / 2   0 cos 2  ,
   0  r   0 sin 2 .
Show that the stresses in 5.2.3.2 are equivalent to a stress 11   0 ,  22  2  0 . It is
easiest to work backwards – start with the stress components in the e1, e2 , e3  basis and
5.2.4.4.
use the basis change formulas to find the stresses in the er , e , e z  basis
Plot a graph showing the variation of hoop stress   /  0 with  at r  a (the surface of
the hole). What is the value of the maximum stress, and where does it occur?
55
p(s)
5.2.5. Find an expression for the vertical displacement u1 ( x2 ) of the
surface of a half-space that is subjected to a distribution of pressure
p(s) as shown in the picture. Show that the slope of the surface can
be calculated as

2
u2 2 1 

x2
E

e2
p( s)ds
 x2  s
e1
L
5.3. Complex variable Solutions to Static Linear Elasticity Problems
5.3.1. A long cylinder is made from an isotropic, linear elastic solid
with shear modulus  . The solid is loaded so
(i) the resultant forces and moments acting on the ends of the
cylinder are zero;
(ii) the body force b  b( x1, x2 )e3 in the interior of the solid acts
parallel to the axis of the cylinder; and (iii) Any tractions
t  t ( x1 , x2 )e3 or displacements u*  u* ( x1 , x2 )e3 imposed on the
sides of the cylinder are parallel to the axis of the cylinder.
Under these conditions, the displacement field at a point far from
the ends of the cylinder has the form u  u ( x1, x2 )e3 , and the solid is said to deform in a state of antiplane shear.
5.3.1.1.
Calculate the strain field in the solid in terms of u.
5.3.1.2.
Find an expression for the nonzero stress components in the solid, in terms of u and
material properties.
5.3.1.3.
Find the equations of equilibrium for the nonzero stress components.
5.3.1.4.
Write down boundary conditions for stress and displacement on the side of the cylinder
5.3.1.5.
Hence, show that the governing equations for u reduce to
2
2



 u  u
u ( x)
  u (x)


b0
u (x)  u* ( x) x  S1

n1 ( x) 
n2 ( x)   t ( x) x  S1 


2
2

x

x
x1 x2
 

1
2





5.3.2. Let ( z ) be an analytic function of a complex number z  x1  ix2 . Let v ( x1 , x2 ) , w( x1, x2 )
denote the real and imaginary parts of ( z ) .
5.3.2.1.
Since ( z ) is analytic, the real and imaginary parts must satisfy the Cauchy Riemann
conditions
v w
w
v


x1 x2
x1
x2
Show that
 2v  2v
2w 2w

0

0
x12 x22
x12 x22
56
5.3.2.2.
Deduce that the displacement and stress in a solid that is free of body force, and loaded on
its boundary so as to induce a state anti-plane shear (see problem 1) can be derived an
analytic function ( z ) , using the representation
2u ( x1, x2 )  ( z )  ( z )
 31  i 32   '( z )
5.3.3.
Calculate the displacements and stresses generated when the complex potential
F
( z )  
log( z )
2
is substituted into the representation described in Problem 2. Show that the solution represents the
displacement and stress in an infinite solid due to a line force acting in the e3 direction at the origin.
5.3.4.
Calculate the displacements and stresses generated when the complex potential
i b
( z )  
log( z )
2
is substituted into the representation described in Problem 2. Show that the solution represents the
displacement and stress due to a screw dislocation in an infinite solid, with burgers vector and line
direction parallel to e3 . (To do this, you need to show that (a) the displacement field has the correct
character; and (b) the resultant force acting on a circular arc surrounding the dislocation is zero)
5.3.5.
Calculate the displacements and stresses generated when the complex potential

a2 
( z )   0  z 



z


is substituted into the representation described in Problem 2. Show that the solution represents the
displacement and stress in an infinite solid, which contains a hole with radius a at the origin, and is
subjected to anti-plane shear at infinity.
5.3.6.
Calculate the displacements and stresses generated when the complex potential
( z )  i 0 z 2  a 2
is substituted into the representation described in Problem 2. Show that the solution represents the
displacement and stress in an infinite solid, which contains a crack with length a at the origin, and is
subjected to a prescribed anti-plane shear stress at infinity. Use the procedure given in Section 5.3.6 to
calculate
z 2  a2
5.3.7. Consider complex potentials ( z )  az  b,  ( z )  cz  d , where a, b, c, d are complex numbers.
Let
E
D  (3  4 )( z )  z ( z )   ( z )
(1  )

11   22  2 ( z)  ( z)


11   22  2i12  2 z ( z)  ( z)

be a displacement and stress field derived from these potentials.
5.3.7.1.
Find values of a,b,c,d that represent a rigid displacement u1  w1   x2 , u2  w2   x1
where w1 , w2 are (real) constants representing a translation, and  is a real constant
representing an infinitesimal rotation.
57
5.3.7.2.
5.3.8.
Find values of a,b,c,d that correspond to a state of uniform stress
Note that the solutions to 7.1 and 7.2 are not unique.
Show that the complex potentials
p a 2  pbb 2
( pa  pb )
( z )  a
z
 ( z)  
2
2
4(b  a )
2(b 2  a 2 ) z
give the stress and displacement field in a pressurized circular cylinder
which deforms in plane strain (it is best to solve this problem using polar
coordinates)
5.3.9.
The complex potentials
E  b1  ib2 
E (b1  ib2 )
( z )  i
log( z )
( z)  i
log( z )
2
8 (1   )
8 (1   2 )
generate the plane strain solution to an edge dislocation at the origin
of an infinite solid. Work through the algebra necessary to determine
the stresses (you can check your answer using the solution given in
Section 5.3.4). Verify that the resultant force exerted by the internal
tractions on a circular surface surrounding the dislocation is zero.
ix2
r

x1
5.3.10. When a stress field acts on a dislocation, the dislocation tends to move through the solid. Formulas
for these forces are derived in Section 5.9.5. For the particular case of a straight edge dislocation, with
burgers vector bi , the force can be calculated as follows:
Let  ijD denote the stress field in an infinite solid containing the dislocation (calculated using the
formulas in Section 5.3.4
Let  ij denote the actual stress field in the solid (including the effects of the dislocation itself, as
well as corrections due to boundaries in the solid, or externally applied fields)
Define  ij   ij   ijD denote the difference between these quantities.
The force can then be calculated as Fi ij 3  jk bk .
Consider two edge dislocations in an infinite solid, each with burgers
vector b1  b, b2  0 . One dislocation is located at the origin, the
other is at position ( x1 , x2 ) . Plot contours of the horizontal
component of force acting on the second dislocation due to the stress
field of the dislocation at the origin. (normalize the force as
8 (1  2 ) F1  x12  x22  / Eb ).


x2
r

x1
58
5.3.11. The figure shows an edge dislocation below the surface of an
elastic solid. Use the solution given in Section 5.3.12, together with
the formula in Problem 5.3.9 to calculate an expression for the force
acting on the dislocation.
x1
h
x2
5.3.12. The figure shows an edge dislocation with burgers vector
b1  b, b2  0 that lies in a strained elastic film with thickness h.

x1
h
The film and substrate have the same elastic moduli. The stress in
the film consists of the stress due to the dislocation, together with a
tensile stress 11   0 .
Calculate the force acting on the
x2
dislocation, and hence find the film thickness for which the
dislocation will be attracted to the free surface and escape from the
film. You will need to use the formula given in problem 5.3.10 to calculate the force on the dislocation.
5.3.13. The figure shows a dislocation in an elastic solid with Young’s
modulus E and Poisson’s ratio  , which is bonded to a rigid solid.
The solution can be generated from complex potentials
( z )  0 ( z )  1 ( z )
 ( z )  0 ( z )  1 ( z )
where
E  b1  ib2 
0 ( z )  i
log( z  ih)
8 (1  2 )
E (b1  ib2 )
x1
h
x2
E  b1  ib2 
h
8 (1  )
8 (1  ) z  ih
is the solution for a dislocation at position z0  ih in an infinite solid, and
0 ( z )  i
2
log( z  ih) 
2


1 ( z )  (3  4 )0 ( z )  z  0 ( z )  z 0 ( z )  0 ( z )  / (3  4 )
1 ( z )  z 0 ( z )  0 ( z ) /  3  4 
corrects the solution to satisfy the zero displacement boundary condition at the interface.
5.3.13.1. Show that the solution satisfies the zero displacement boundary condition
5.3.13.2. Calculate the force acting on the dislocation, in terms of h and relevant material properties.
You will need to use the formula from 5.3.10 to calculate the force on the dislocation.
5.3.13.3. Calculate the distribution of stress along the interface between the elastic and rigid solids,
in terms of h and x1
59
5.3.14. The figure shows a rigid cylindrical inclusion with radius a
embedded in an isotropic elastic matrix. The solid is subjected to
e2
a uniform uniaxial stress 11   0 at infinity. The goal of this
problem is to calculate the stress fields in the matrix.
r
5.3.14.1. Write down the boundary conditions on the

a
displacement field at r=a
e1
5.3.14.2. Show that the boundary conditions can be satisfied by
complex potentials of the form
   a2 
 
 a2 a4 
( z )  0  z 
( z)   0  z 



4 
z 
2 
z
z 3 
where  ,  ,  are three real valued coefficients whose values you will need to determine. The algebra
in this problem can be simplified by noting that z  a2 / z on the boundary of the inclusion.
5.3.14.3. Find an expression for the stresses acting at the inclusion/matrix boundary
5.3.14.4. The interface between inclusion and matrix fails when the normal stress acting on the
interface reaches a critical stress  crit . Find an expression for the maximum tensile stress
that can be applied to the material without causing failure.
5.3.15. The figure shows a cylindrical inclusion with radius a and
Young’s modulus and Poisson’s ratio E p , p embedded in an
e2
isotropic elastic matrix with elastic constants E, . The solid is
r
subjected to a uniform uniaxial stress 11   0 at infinity. The

a
goal of this problem is to calculate the stress fields in both the
e1
particle and the matrix. The analylsis can be simplified greatly by
assuming a priori that the stress in the particle is uniform (this is
not obvious, but can be checked after the full solution has been
obtained). Assume, therefore, that the stress in the inclusion is
11  p11,  22  p22 s12  0 , where p11 , p22 are to be
determined. The solution inside the particle can therefore be derived from complex potentials
p  p22
p p
 p ( z )  11
z
 p ( z)  22 11 z
4
2
In addition, assume that the solution in the matrix can be derived from complex potentials of the form
   a2 
0 
 a2 a4 
( z )  0  z 

(
z
)


z





4 
z 
2 
z
z 3 
where  ,  ,  are three real valued coefficients whose values you will need to determine.
5.3.15.1. Write down the boundary conditions on the displacement field at r=a. Express this
boundary condition as an equation relating  p ,  p to ,  in terms of material properties.
5.3.15.2.
Write down boundary conditions on the stress components  rr , r at r=a. Express this
boundary condition as an equation relating  p ,  p to ,  .
5.3.15.3.
5.3.15.4.
Calculate expressions for p11 , p22 ,  ,  ,  in terms of  0 and material properties.
The inclusion fractures when the maximum principal stress acting in the particle reaches a
critical stress  crit . Find an expression for the maximum tensile stress that can be applied
to the material without causing failure.
60
5.3.16. The figure shows a slit crack in an infinite solid. Using the
solution given in Section 5.3.6, calculate the stress field very near
the right hand crack tip (i.e. find the stresses in the limit as
r1 / a  0 ). Show that the results are consistent with the asymptotic
crack tip field given in Section 5.2.9, and deduce an expression for
the crack tip stress intensity factors in terms of


 22
, 12
e2
r2


a
a
r
r1

e1
and a.
5.3.17. Two identical cylindrical roller bearings with radius 1cm are pressed into
contact by a force P per unit out of plane length as indicated in the figure. The
bearings are made from 52100 steel with a uniaxial tensile yield stress of 2.8GPa.
Calculate the force (per unit length) that will just initiate yield in the bearings,
and calculate the width of the contact strip between the bearings at this load.
P
P
5.3.18. The figure shows a pair of identical involute spur gears. The
contact between the two gears can be idealized as a line contact
between two cylindrical surfaces. The goal of this problem is to
find an expression for the maximum torque Q that can be
transmitted through the gears. The gears can be idealized as
isotropic, linear elastic solids with Young’s modulus E and
Poisson’s ratio  .
As a representative configuration, consider the instant when a single pair of
gear teeth make contact exactly at the pitch point. At this time, the geometry
can be idealized as contact between two cylinders, with radius r  R p sin  ,
where R p is the pitch circle radius of the gears and  is the pressure angle.
The cylinders are pressed into contact by a force P  Q /( R p cos  ) .
5.3.18.1.
Find a formula for the area of contact between the two gear teeth,
in terms of Q , R p ,  , b and representative material properties.
5.3.18.2.
Find a formula for the maximum contact pressure acting on the
contact area, in terms of Q , R p ,  , b and representative material properties.
5.3.18.3.
Suppose that the gears have uniaxial tensile yield stress Y. Find a formula for the critical
value of Q required to initiate yield in the gears.
61
5.4. Solutions to 3D static problems in linear elasticity
5.4.1.
Consider the Papkovich-Neuber potentials
(1  ) 0
 (1  ) 0
i 
x3i3

(3x32  R 2 )
(1   )
(1   )
5.4.1.1.
Verify that the potentials satisfy the equilibrium equations
5.4.1.2.
Show that the fields generated from the potentials correspond to a state of uniaxial stress,
with magnitude  0 acting parallel to the e3 direction of an infinite solid
5.4.2.
Consider the fields derived from the Papkovich-Neuber potentials
(1  ) p
2 (1  ) p 2
i 
xi

R
(1   )
(1   )
5.4.2.1.
Verify that the potentials satisfy the equilibrium equations
5.4.2.2.
Show that the fields generated from the potentials correspond to a state of hydrostatic
tension  ij  pij
5.4.3.
Consider the Papkovich-Neuber potentials
5.4.3.1.
5.4.3.2.
5.4.3.3.
x
3
 i   xi   i
   R2 
3
R
R
Verify that the potentials satisfy the governing equations
Show that the potentials generate a spherically symmetric displacement field
Calculate values of  and  that generate the solution to an internally pressurized
spherical shell, with pressure p acting at R=a and with surface at R=b traction free.
5.4.4.
Verify that the Papkovich-Neuber potential
P
i  i
 0
4 R
generates the fields for a point force P  P1e1  P2e2  P3e3 acting at the origin of a large (infinite)
elastic solid with Young’s modulus E and Poisson’s ratio  . To this end:
5.4.4.1.
Verify that the potentials satisfy the governing equation
5.4.4.2.
Calculate the stresses
5.4.4.3.
Consider a spherical region with radius R surrounding the origin. Calculate the resultant
force exerted by the stress on the outer surface of this sphere, and show that they are in
equilibrium with a force P.
5.4.5. Consider an infinite, isotropic, linear elastic solid with Young’s modulus E and Poisson’s ratio  .
Suppose that the solid contains a rigid spherical particle (an inclusion) with radius a and center at the
origin. The particle is perfectly bonded to the elastic matrix, so that ui  0 at the particle/matrix
interface. The solid is subjected to a uniaxial tensile stress  33   0 at infinity. Calculate the stress
field in the elastic solid. To proceed, note that the potentials
(1  ) 0
 (1  ) 0
i 
x3i3

(3x32  R 2 )
(1   )
(1   )
generate a uniform, uniaxial stress  33   0 (see problem 1). The potentials
62
a3 pijT 
xi x j 
(5R 2  a 2 )ij  3a 2


3R3
15R3 
R2 
are a special case of the Eshelby problem described in Section 5.4.6, and generate the stresses outside a
i 
T
a3 pik
xk

spherical inclusion, which is subjected to a uniform transformation strain. Let pijT  Aij  Bi 3 j 3 ,
where A and B are constants to be determined. The two pairs of potentials can be superposed to
generate the required solution.
5.4.6. Consider an infinite, isotropic, linear elastic solid with Young’s modulus E and Poisson’s ratio  .
Suppose that the solid contains a spherical particle (an inclusion) with radius a and center at the origin.
The particle has Young’s modulus E p and Poisson’s ratio  p , and is perfectly bonded to the matrix, so
that the displacement and radial stress are equal in both particle and matrix at the particle/matrix
interface. The solid is subjected to a uniaxial tensile stress  33   0 at infinity. The objective of this
problem is to calculate the stress field in the elastic inclusion.
5.4.6.1.
Assume that the stress field inside the inclusion is given by  ij  A 0 ij  B 0 i3 j 3 .
5.4.6.2.
i 
Calculate the displacement field in the inclusion (assume that the displacement and rotation
of the solid vanish at the origin).
The stress field outside the inclusion can be generated from Papkovich-Neuber potentials
T
(1  ) 0
a3 pik
xk
x3i3 
(1   )
3R3
3 T

a pij
 (1  ) 0
(3x32  R 2 ) 
(1   )
15R3

2
2
2 xi x j 
 (5R  a )ij  3a

R2 

where pijT  C 0ij  D 0i3 j 3 , and C and D are constants to be determined.
5.4.6.3.
5.4.6.4.
Use the conditions at r=a to find expressions for A,B,C,D in terms of geometric and
material properties.
Hence, find the stress field inside the inclusion.
5.4.7. Consider the Eshelby inclusion problem described in Section 5.4.6. An infinite homogeneous, stress
free, linear elastic solid has Young’s modulus E and Poisson’s ratio  . The solid is initially stress free.
An inelastic strain distribution  ijT
is introduced into an ellipsoidal region of the solid B (e.g. due to
thermal expansion, or a phase transformation). Let ui denote the displacement field,  ij   ije   ijT
denote the total strain distribution, and let  ij denote the stress field in the solid.
5.4.7.1.
Write down an expression for the total strain energy  I within the ellipsoidal region, in
terms of  ij ,  ij and  ijT .
5.4.7.2.
Write down an expression for the total strain energy outside the ellipsoidal region,
expressing your answer as a volume integral in terms of  ij and  ij . Using the divergence
theorem, show that the result can also be expressed as
1
O    ij n j ui dA
2

S
where S denotes the surface of the ellipsoid, and n j are the components of an outward unit
vector normal to B. Note that, when applying the divergence theorem, you need to show
that the integral taken over the (arbitrary) boundary of the solid at infinity does not
contribute to the energy – you can do this by using the asymptotic formula given in Section
5.4.6 for the displacements far from an Eshelby inclusion.
63
5.4.7.3.
The Eshelby solution shows that the strain  ij   ije   ijT inside B is uniform. Write down
the displacement field inside the ellipsoidal region, in terms of  ij (take the displacement
and rotation of the solid at the origin to be zero). Hence, show that the result of 7.2 can be
re-written as
1
O    ij  ik xk n j dA
2

S
5.4.7.4.
Finally, use the results of 7.1 and 7.3, together with the divergence theorem, to show that
the total strain energy of the solid can be calculated as
1
  O   I     ij  ijT dV
2
B
5.4.8.
Using the solution to Problem 7, calculate the total strain energy of an initially stress-free isotropic,
linear elastic solid with Young’s modulus E and Poisson’s ratio  , after an inelastic strain  ijT is
introduced into a spherical region with radius a in the solid.
5.4.9. A steel ball-bearing with radius 1cm is pushed into a flat steel surface by a force P. Neglect friction
between the contacting surfaces. Typical ball-bearing steels have uniaxial tensile yield stress of order
2.8 GPa. Calculate the maximum load that the ball-bearing can withstand without causing yield, and
calculate the radius of contact and maximum contact pressure at this load.
5.4.10. The contact between the wheel of a locomotive and the head of a rail may
be approximated as the (frictionless) contact between two cylinders, with
identical radius R as illustrated in the figure. The rail and wheel can be
idealized as elastic-perfectly plastic solids with identical Young’s modulus E,
Poisson’s ratio  and yield stress Y. Find expressions for the radius of the
contact patch, the contact area, and the contact pressure as a function of the
load acting on the wheel and relevant geometric and material properties. By
estimating values for relevant quantities, calculate the maximum load that can
be applied to the wheel without causing the rail to yield.
5.4.11. The figure shows a rolling element bearing. The inner raceway has
radius R, and the balls have radius r, and both inner and outer raceways
are designed so that the area of contact between the ball and the
raceway is circular. The balls are equally spaced circumferentially
around the ring. The bearing is free of stress when unloaded. The
bearing is then subjected to a force P as shown. This load is transmitted
through the bearings at the contacts between the raceways and the balls
marked A, B, C in the figure (the remaining balls lose contact with the
raceways but are held in place by a cage, which is not shown). Assume
that the entire assembly is made from an elastic material with Young’s
64
modulus E and Poisson’s ratio 
5.4.11.1. Assume that the load causes the center of the inner raceway to move vertically upwards by
a distance  , while the outer raceway remains fixed. Write down the change in the gap
between inner and outer raceway at A, B, C, in terms of 
5.4.11.2. Hence, calculate the resultant contact forces between the balls at A, B, C and the raceways,
in terms of  and relevant geometrical and material properties.
5.4.11.3. Finally, calculate the contact forces in terms of P
5.4.11.4. If the materials have uniaxial tensile yield stress Y, find an expression for the maximum
force P that the bearing can withstand before yielding.
P
5.4.12. A rigid, conical indenter with apex angle
2  is pressed into the surface of an isotropic,
linear elastic solid with Young’s modulus E

and Poisson’s ratio  .
5.4.12.1. Write down the initial gap between
r
a
the two surfaces g (r )
5.4.12.2. Find the relationship between the
depth of penetration h of the indenter and the radius of contact a
5.4.12.3. Find the relationship between the force applied to the contact and the radius of contact, and
hence deduce the relationship between penetration depth and force. Verify that the contact
dP
stiffness is given by
 2E*a
dh
5.4.12.4. Calculate the distribution of contact pressure that acts between the contacting surfaces.
5.4.13. A sphere, which has radius R, is dropped from height h onto the flat surface of a large solid. The
sphere has mass density  , and both the sphere and the surface can be idealized as linear elastic solids,
with Young’s modulus E and Poisson’s ratio  . As a rough approximation, the impact can be
idealized as a quasi-static elastic indentation.
5.4.13.1. Write down the relationship between the force P acting on the sphere and the displacement
of the center of the sphere below x2  R
5.4.13.2. Calculate the maximum vertical displacement of the sphere below the point of initial
contact.
5.4.13.3. Deduce the maximum force and contact pressure acting on the sphere
5.4.13.4. Suppose that the two solids have yield stress in uniaxial tension Y. Find an expression for
the critical value of h which will cause the solids to yield
5.4.13.5. Calculate a value of h if the materials are steel, and the sphere has a 1 cm radius.
65
5.5. Solutions to generalized plane problems for anisotropic linear elastic solids
5.5.1. Consider a plane, anisotropic elastic solid, which
is loaded so as to induce an anti-plane shear
deformation field of the form u  u3 ( x1, x2 ) e3 . Find
the three equations of equilibrium in terms of u3
and relevant elastic constants. Show that, since
the elastic constants must be positive, an antiplane shear deformation field can only satisfy the
equilibrium equations if the elastic constants
satisfy
S
e2
e3
b
R
t
Deformed
Configuration
e1
c15  c46  c14  c56  c24  c25  0
and that under these conditions the equilibrium equation reduces to
c55u3,11  c44u3 ,22 2c45u3 ,12  0
5.5.2. Consider a displacement field of the form u3  Re( f ( z )) , where f(z) is an analytic function, and
z  x1  px2 .
5.5.2.1.
5.5.2.2.
5.5.2.3.

Show that u3 ,11  Re  f ''( z )
u3 ,12  Re  pf ''( z ) u3 ,22  Re p 2 f ''( z )

Hence, show that the governing equation of problem 5.5.1 can be satisfied by
setting u3  Re( f ( z )) , z  x1  px2 , with p a complex number given by
i  c45
2
p
  c44c55  c45
c44
Show that the nonzero stresses can be computed from the expression
 31  i 32 

2
(1  ip) f '( z)  (1  ip) f '( z)
5.5.3. Show that the analytic function f ( z )  ( F3 / 2 )log( z ) ,
substituted in the representation of problem 5.5.1 and 5.5.2 generates
the solution to a point force acting in the e3 direction at the origin of
an infinite solid. To do this, you need to show (1) that the solution
generates a single valued displacement field, and (2) that the resultant
force exerted by the tractions acting on a circular arc surrounding the
origin is in equilibrium with the point force.
ix2
r
F3

x1
5.5.4. Guided by 5.5.3, construct the solution to a straight screw
dislocation in an anisotropic elastic solid, and calculate the stress field.
5.5.5. Calculate numerical values for the Stroh matrices A and B for (a) Cu, and (b) Al, assuming that
<100> directions are parallel to the coordinate axes.
5.5.6. Calculate numerical values for the Barnett-Lothe tensors S, H and L for copper with <100>
directions parallel to the coordinate axes. Find the tensors using two approaches: (i) by substituting
into the formulas in terms of A and B, and (ii) by evaluating the integral formulas in 5.5.11.
66
5.5.7. Calculate an expression for the inverse of the impedance tensor for a cubic material with <100>
directions parallel to the coordinate axes (the expression is simpler than the formula for the impedance
tensor itself)
5.5.8.
Find an expression for the fundamental elasticity matrix N for a cubic material with <100>
directions parallel to the coordinate axes. Verify that pi are the eigenvalues of N, and that [a, b]T are
its eigenvectors.
5.5.9. Let N1 , N 2 be the sub-matrices of the fundamental elasticity tensor defined in Section 5.5.6, let A
and B denote the matrices of Stroh eigenvectors, and let P  diag( p1, p2 , p3 ) be the diagonal matrix of
Stroh eighevalues. Show that
AP  N1A  N 2B
5.5.10. The Stroh representation for a uniform state of generalized plane strain in an anisotropic solid is
u  2 Re( AZq) φ  2 Re(BZq)
Z  diag ( z1 , z2 , z3 )
q  AT t 2  BT ε1
zi  x1  pi x2
where
5.5.10.1.
5.5.10.2.
5.5.10.3.
 11 
12 


ε1   12 
t 2   22 
 2 31 
 32 
Use the properties of the Stroh matrices listed in Section 5.5.9 to show that
u
φ
 ε1
 t2
x1
x1
Show also that
u
φ
 N2t 2  N1ε1
 N1T t 2  N3ε1
x2
x2
Hence, deduce that
ε 2  N 2t 2  N1ε1

t1   N1T t 2  N3ε1

where
 12 
ε 2    22 
 2 32 
11 
t1  12 
 31 
5.5.11. Use the solution given in Section 5.5.13 to calculate the stress distribution due to an edge
dislocation with burgers vector b  be1 in a Cu crystal, with <100> directions parallel to the coordinate
directions. Plot contours of the radial and hoop stresses around the dislocation, and compare the results
with those given in Section 5.3.4.
5.5.12. Use the solution given in Section 5.5.14 to calculate the force acting on a dislocation near
the free surface of a single crystal of Cu. Assume that the dislocation has burgers vector
b  be1 , and that the <100> directions of the crystal are parallel to the coordinate axes.
67
5.6. Solutions to dynamic problems for isotropic linear elastic solids
  A sin( pi xi  cL t ) , where pi is a constant unit vector
5.6.1. Consider the Love potentials  i  0
and A is a constant.
5.6.1.1.
Verify that the potentials satisfy the appropriate governing equations
5.6.1.2.
Calculate the stresses and displacements generated from these potentials.
5.6.1.3.
Briefly, interpret the wave motion represented by this solution.
  0 , where pi is a constant unit vector
5.6.2. Consider the Love potentials  i  U i sin( pk xk  cs t )
and U i is a constant unit vector.
5.6.2.1.
Find a condition relating U i and pi that must be satisfied for this to be a solution to the
governing equations
5.6.2.2.
Calculate the stresses and displacements generated from these potentials.
5.6.2.3.
Briefly, interpret the wave motion represented by this solution.
1
f (t  R / cL ) satisfy the governing equations for Love potentials. Find
R
expressions for the corresponding displacement and stress fields.
5.6.3.
Show that i  0

5.6.4. Calculate the radial distribution of Von-Mises effective stress surrounding a spherical cavity of
radius a, which has pressure p0 suddenly applied to its surface at time t=0. Hence, find the location in
the solid that is subjected to the largest Von-Mises stress, and the time at which the maximum occurs.
5.6.5.
Calculate the displacement and stress fields generated by the Love potentials
A
i  0
  sin   t  ( R  a) / cL    
R
Calculate the traction acting on the surface at r=a. Hence, find the Love potential that generates the
fields around a spherical cavity with radius a, which is subjected to a harmonic pressure
p(t )  p0 sin t . Plot the amplitude of the surface displacement at r=a (normalized by a) as a function
of  a / cL .
5.6.6. Calculate the distribution of kinetic and potential energy near the surface of a half-space that
contains a Rayleigh wave with displacement amplitude U 0 and wave number k. Take Poisson’s ratio
  0.3 . Calculate the total energy per unit area of the wave (find the total energy in one wavelength,
then divide by the wavelength). Estimate the energy per unit area in Rayleigh waves associated with
earthquakes.
5.6.7. The figure shows a surface-acoustic-wave device
d
that is intended to act as a narrow band-pass filter. A
piezoelectric substrate has two transducers attached to
its surface – one acts as an “input” transducer and the
other as “output.” The transducers are electrodes: a
charge can be applied to the input transducer; or
detected on the output. Applying a charge to the input
Output
Input
transducer induces a strain on the surface of the
substrate: at an appropriate frequency, this will excite a Rayleigh wave in the solid. The wave
propagates to the “output” electrodes, and the resulting deformation of the substrate induces a charge
68
that can be detected. If the electrodes have spacing d, calculate the frequency at which the surface will
be excited. Estimate the spacing required for a 1GHz filter made from AlN with Young’s modulus 345
GPa and Poisson’s ratio 0.3
5.6.8. The figure shows a thin elastic strip, which is bonded to rigid
solids on both its surfaces. The strip has shear modulus  and
wave speed cs , and acts as a wave-guide. The goal of this
 ,cs
problem is to calculate the displacement field associated with
transverse wave propagation down the strip.
5.6.8.1.
Assume that the displacement has the form
u3  f ( x2 )exp(ik ( x1  ct ))
By substituting into the Cauchy-Navier equation, show that
 c2

d2
f ( x2 )  k 2   1 f ( x2 )  0
 c2

dx22
 s

Hence, write down the general solution for f ( x2 )
5.6.8.2.
Show that the boundary conditions admit solutions of the form
A sin(n x2 / H )

f ( x2 )  
 B cos(( / 2  n ) x2 / H )
where n is an integer, so that

e2
e1
H
H

u3  U 0 cos 2 ( n / 2)sin(n x2 / 2 H )  sin 2 ( n / 2)cos(n x2 / 2 H ) exp(ik ( x1  ct ))
5.6.8.3.
5.6.8.4.
Find an expression for the phase velocity of the wave, and plot the phase velocity as a
function of kH/n.
Calculate the dispersion relation for the wave and hence deduce an expression for the group
velocity. Plot the group velocity as a function of kH/n.
5.6.9. Consider the Love wave described in Section 5.6.4.
5.6.9.1.
Consider first a system with    f , cs / csf  2 , as discussed at the end of 5.6.4. Find an
5.6.9.2.
expression for the group velocity of the wave, and plot a graph showing the group velocity
(normalized by shear wave speed in the layer) as a function of kH.
Consider a system with    f , cs / csf  1/ 2 . Plot graphs showing both the phase
velocity and the group velocity in the layer as a function of kH.
5.6.10. In this problem, you will investigate the energy associated
e2
with wave propagation down a simple wave guide. Consider an
isotropic, linear elastic strip, with thickness 2H, shear modulus
H
 and wave speed cs as indicated in the figure. The solution
 ,cs
e1
H
for a wave propagating in the e1 direction, with particle
velocity u  u3e3 is given in Section 5.6.6 of the text.
5.6.10.1. The flux of energy associated with wave propagation along a wave-guide can be computed
from the work done by the tractions acting on an internal material surface. The work done
per cycle is given by
69
P 
5.6.10.2.
1
T
T H
   ij n j
0 H
where T is the period of oscillation and n j   j1 is a unit vector normal to an internal
plane perpendicular to the direction of wave propagation. Calculate P for the nth wave
propagation mode.
The average kinetic energy of a generic cross-section of the wave-guide can be calculated
from
1
K 
T
5.6.10.3.
dui
dx2 dt
dt
T H
 dui dui
0 H
2 dt dt

dx2 dt
Find K for nth wave propagation mode.
The average potential energy of a generic cross-section of the wave-guide can be calculated
from
1
 
T
T H

0 H
1
 ij  ij dx2 dt
2
Find  for nth wave propagation mode. Check that K = 
5.6.10.4.
The speed of energy flux down the wave-guide is defined as ce  P /  K    . Find
ce for the nth propagation mode, and compare the solution with the expression for the
group velocity of the wave
cs kH
d
cg 

dk
(n / 2) 2  k 2 H 2
5.7. Energy methods for solving static linear elasticity problems
5.7.1. A shaft with length L and square cross section is fixed at one
end, and subjected to a twisting moment T at the other. The shaft is
made from a linear elastic solid with Young’s modulus E and
Poisson’s ratio  . The torque causes the top end of the shaft to
rotate through an angle  .
5.7.1.1.
Consider the following displacement field
v1  
5.7.1.2.
5.7.1.3.
5.7.1.4.

x1x3
v2 

x2 x3
v3  0
T
a
a
L

e3
L
L
Show that this is a kinematically admissible
e2
displacement field for the twisted shaft.
e1
Calculate the strains associated with this kinematically
admissible displacement field
Hence, show that the potential energy of the shaft is
You may assume that the potential energy of the torsional load is T
Find the value of  that minimizes the potential energy, and hence estimate the torsional
stiffness of the shaft.
70
5.7.2. In this problem you will use the principle of minimum potential
energy to find an approximate solution to the displacement in a
pressurized cylinder. Assume that the cylinder is an isotropic, linear
elastic solid with Young’s modulus E and Poisson’s ratio  , and
subjected to internal pressure p at r=a.
5.7.2.1.
Approximate the radial displacement field as ur  C1  C2 r ,
where C1 , C2 are constants to be determined. Assume all
other components of displacement are zero. Calculate the
strains in the solid  rr , 
5.7.2.2.
Find an expression for the total strain energy of the cylinder
per unit length, in terms of C1 , C2 and relevant geometric
and material parameters
5.7.2.3.
Hence, write down the potential energy (per unit length) of the cylinder.
5.7.2.4.
Find the values of C1 , C2 that minimize the potential energy
5.7.2.5.


Plot a graph showing the normalized radial displacement field E b 2  a 2 ur (r ) /( pab 2 ) as
a function of the normalized position (r  a) /(b  a) in the cylinder, for   0.3 , and
b / a  1.1 and b / a  2 . On the same graph, plot the exact solution, given in Section 4.1.9.
5.7.3. A bi-metallic strip is made by welding together two materials
with identical Young’s modulus and Poisson’s ratio E, , but with

h
different thermal expansion coefficients 1 ,  2 , as shown in the

h
picture. At some arbitrary temperature T0 the strip is straight and
free of stress. The temperature is then increased to a new value T ,
causing the strip to bend. Assume that, after heating, the displacement field in the strip can be
u2   x12 /(2 R )   x2 u3  0 , where  , R,  are constants to
approximated as u1   x1  x1x2 / R
be determined.
5.7.3.1.
Briefly describe the physical significance of the shape changes associated with  , R,  .
5.7.3.2.
Calculate the distribution of (infinitesimal) strain associated with the kinematically
admissible displacement field
5.7.3.3.
Hence, calculate the strain energy density distribution in the solid. Don’t forget to account
for the effects of thermal expansion
5.7.3.4.
Minimize the potential energy to determine values for  , R,  in terms of relevant
geometric and material parameters.
5.7.4. By guessing the deflected shape, estimate the
P
stiffness of a clamped—clamped beam subjected to a
point force at mid-span. Note that your guess for the
E, I
dw
deflected shape must satisfy w( x1 ) 
 0 , so you
dx1
L
can’t assume that it bends into a circular shape as done in
class.
Instead, try a deflection of the form
w( x1 )  1  cos  2 x1 / L  , or a similar function of your choice (you could try a suitable polynomial, for
example). If you try more than one guess and want to know which one gives the best result, remember
71
that energy minimization always overestimates stiffness. The best guess is the one that gives the lowest
stiffness.
5.7.5. A slender rod with length L and cross sectional area A is subjected to an axial
body force b  b( x2 )e2 . Our objective is to determine an approximate solution to
the displacement field in the rod.
5.7.5.1.
Assume that the displacement field has the form
u2  w( x2 ) u1  u3  0
where the function w is to be determined. Find an expression for the
strains in terms of w and hence deduce the strain energy density.
5.7.5.2.
Show that the potential energy of the rod is
L
b(x2)
e2
2
L 
L
e1
1 
1  dw 

V ( w)  EA
dx

A
b( x2 )w( x2 )dx2

 2
1  2 1    0 2  dx2 
0
5.7.5.3.
To minimize the potential energy, suppose that w is perturbed from the value the minimizes
V to a value w   w . Assume that  w is kinematically admissible, which requires that
 w  0 at any point on the bar where the value of w is prescribed. Calculate the potential
energy V ( w   w) and show that it can be expressed in the form
1
V ( w   w)  V ( w)   V   2V
2
where V is a function of w only,  V is a function of w and  w , and  2V is a function of
 w only.
5.7.5.4.
As discussed in Section 8 of the online notes (or in class), if V is stationary at  w  0 ,
then  V  0 . Show that, to satisfy  V  0 , we must choose w to satisfy


L

5.7.5.5.
L
EA(1  )
dw d w
dx2  A b wdx2  0
(1  2 )(1   ) dx2 dx2
0
0
Integrate the first term by parts to deduce that, to minimize, V, w must satisfy

d 2w
 b( x2 )  0
dx22
Show that this is equivalent to the equilibrium condition
d 22
 b2  0
dx2
Furthermore, deduce that if w is not prescribed at either x2  0, x2  L or
both, then the boundary conditions on the end(s) of the rod must be
dw
0
dx2
Show that this corresponds to the condition that  22  0 at a free end.
5.7.5.6.
Use your results in (2.5) to estimate the displacement field in a bar
with mass density  , which is attached to a rigid wall at x2  L , is
free at x2  0 , and subjected to the force of gravity (acting vertically
downwards…)
L
e2
e1
72
5.8. The Reciprocal Theorem and applications
5.8.1. A planet that deforms under its own gravitational force can be idealized as a
linear elastic sphere with radius a, Young’s modulus E and Poisson’s ratio  that
is subjected to a radial gravitational force b  ( g R / a)e R , where g is the
acceleration due to gravity at the surface of the sphere, and R is the radial
coordinate. Use the reciprocal theorem, together with a hydrostatic stress
distribution  ij  pij as the reference solution, to calculate the change in volume
of the sphere, and hence deduce the radial displacement of its surface.
5.8.2. Consider an isotropic, linear elastic solid with Young’s modulus E, mass density  , and Poisson’s
ratio  , which is subjected to a body force distribution bi per unit mass, and tractions ti on its exterior
surface. By using the reciprocal theorem, together with a state of uniform stress  ij* as the reference
solution, show that the average strains in the solid can be calculated from
1
1 

1 

 ij dV 
x j ti dA 
ij xk tk dA 
x j bi dV 
ij xk bk dV




V
EV
EV
EV
EV 
V
A
A
V
V
5.8.3. A cylinder with arbitrary cross-section rests on a flat surface, and is subjected to a vertical
gravitational body force b    ge3 , where e3 is a unit vector normal to the surface. The cylinder is a
linear elastic solid with Young’s modulus E, mass density  , and Poisson’s ratio  . Define the change
in length of the cylinder as
1
 L   u3 ( x1, x2 , L)dA
A
A
where u3 ( x1 , x2 , L) denotes the displacement of the end x3  L of the cylinder. Show that
 L  W / 2 EA , where W is the weight of the cylinder, and A its cross-sectional area.
5.8.4.
In this problem, you will calculate an expression for the change in potential energy that occurs
when an inelastic strain  ijT is introduced into some part B of an
elastic solid. The inelastic strain can be visualized as a generalized
version of the Eshelby inclusion problem – it could occur as a result
of thermal expansion, a phase transformation in the solid, or plastic
flow. Note that B need not be ellipsoidal.
The figure illustrates the solid of interest. Assume that:
The solid has elastic constants Cijkl
No body forces act on the solid (for simplicity)
Part of the surface of the solid S1 is subjected to a prescribed
S
B
R
t
Deformed
Configuration
displacement ui*
The remainder of the surface of the solid S 2 is subjected to a prescribed traction ti*
73
Let ui0 ,  ij0 , ij0 denote the displacement, strain, and stress in the solid before the inelastic strain is
introduced. Let V0 denote the potential energy of the solid in this state.
Next, suppose that some external process introduces an inelastic strain  ijT into part of the solid. Let
ui ,  ij ,  ij denote the change in stress in the solid resulting from the inelastic strain. Note that
these fields satisfy
The strain-displacement relation  ij  (ui / x j  ui / x j ) / 2
T
The stress-strain law  ij  Cijkl ( kl   kl
) in B, and  ij  Cijkl  kl outside B
Boundary conditions ui  0 on S1 , and  ij n j  0 on S 2 .
5.8.4.1.
Write down an expressions for V0 in terms of ui0 ,  ij0 , ij0
5.8.4.2.
Suppose that ui0 ,  ij0 , ij0 are all zero (i.e. the solid is initially stress free). Write down the
potential energy VS due to ui ,  ij ,  ij .
This is called the “self energy” of the
eigenstrain – the energy cost of introducing the eigenstrain  ijT into a stress-free solid.
5.8.4.3.
Show that the expression for the self-energy can be simplified to
1
VS  
 ij  ijT dV
2

B
5.8.4.4.
Now suppose that ui0 ,  ij0 , ij0 are all nonzero. Write down the total potential energy of the
5.8.4.5.
system VTOT , in terms of ui0 ,  ij0 , ij0 and ui ,  ij ,  ij .
Finally, show that the total potential energy of the system can be expressed as

VTOT  V0  VS   ij0 ijT dV
B
Here, the last term is called the “interaction energy” of the eigenstrain with the applied
load. The steps in this derivation are very similar to the derivation of the reciprocal
theorem.
5.8.5. An infinite, isotropic, linear elastic solid with Young’s modulus E and Poisson’s ratio  is
subjected to a uniaxial tensile stress  0 . As a result of a phase transformation, a uniform dilatational
strain  ijT  ij is then induced in a spherical region of the solid with radius a.
5.8.5.1.
5.8.5.2.
Using the solution to problem 1, and the Eshelby solution, find an expression for the change
in potential energy of the solid, in terms of  , 0 and relevant geometric and material
parameters.
Assume that the interface between the transformed material an the matrix has an energy per
unit area  . Find an expression for the critical stress at which the total energy of the
system (elastic potential energy + interface energy) is decreased as a result of the
transformation
74
5.9. Energetics of Dislocations
5.9.1. Calculate the stress induced by a straight screw dislocation in an infinite solid using the formula in
Section 5.8.4. Compare the solution with the result of the calculation in Problem 5.3.4.
5.9.2. The figure shows two nearby straight screw dislocations in an
infinite solid, with line direction perpendicular to the plane of the figure.
The screw dislocations can be introduced into the solid by cutting the
plane between the dislocations and displacing the upper of the surfaces
e2
d/2
_
S
d/2
e1
S+
created by the cut ( S  ) by be3 / 2 , and the lower ( S  ) by be3 / 2 ,
m
and re-connecting the surfaces. The solid deforms in anti-plane shear,
with a displacement field of the form u  u ( x1, x2 )e3
5.9.2.1.
Write down nonzero components of stress and strain in the solid
5.9.2.2.
Show that the total strain energy of the solid (per unit out of plane distance) can be
expressed as
1
u
U
 3
dA
2
x

A
5.9.2.3.
Show that the potential energy can be re-written as
U 
1
2
d /2

 32 ( x1 )bdx1
 d /2
where the integral is taken along the line x2  0 .
5.9.2.4.
Use the solution for a screw dislocation given in Problem 5.3.4 (or 5.9.1) to show that the
energy can be calculated as
5.9.2.5.
5.9.2.6.
d /2


1
1


 dx1
x1  d / 2 d / 2  x1 

 d /2
Note that the integral is unbounded, as expected. Calculate a bounded expression by
truncating the integral at d / 2   and d / 2  
Calculate the force exerted on one dislocation by the other by differentiating the expression
for the energy. Is the force attractive or repulsive?
Check your answer using the Peach-Koehler formula.
b2
U
4

5.9.3. Calculate the stress induced by an edge dislocation in an infinite solid using the formula in Section
5.8.4. Compare the solution with the result given in 5.3.4
Calculate the nonsingular stress  ij(  ) induced by a screw dislocation in an infinite solid using the
formula in Section 5.9.2. Compare the solution with the result of the calculation in Problem 5.3.4.
5.9.4.
75
5.9.5. Calculate the nonsingular self- energy per unit length of a straight dislocation, using the approach
discussed in Section 5.9.2. (To do this, you have to calculate the energy of a dislocation segment with
finite length, then take the limit of the energy per unit length as the dislocation length goes to infinity).
5.9.6. Calculate the self-energy of a square prismatic dislocation loop with
side length L. Use nonsingular dislocation theory, and give your answer to
zeroth order in the parameter 
e2

L
b
5.9.7. Suppose that the dislocation loop described in the preceding problem is
subjected to a uniaxial tensile stress  ij   0i3 j 3 . Calculate the total
e1
potential energy of the system.
Display your result as a graph of normalized potential energy
D
2
2
V (1   ) / ELb as a function of L/b, for various values of  0 (1   2 ) / E . Take   b / 4 as a
representative value. Hence, estimate (i) an expression for the activation energy required for
homogeneous nucleation of a prismatic dislocation loop, as a function of  0 (1   2 ) / E ; and (ii) the
critical size required for a pre-existing dislocation loop to grow, as a
function of  0 (1   2 ) / E .
5.9.8. Calculate the self-energy of a rectangular glide dislocation loop
with burgers vector b  be1 and side lengths a,d. Use nonsingular
dislocation theory, and give your answer to zeroth order in the
parameter  .
e2

d
b
e1
a
5.9.9. A composite material is made by sandwiching
thin layers of a ductile metal between layers of a
h
hard ceramic. Both the metal and the ceramic

have identical Young’s modulus E and Poisson’s
b
ratio  . The figure shows one of the metal layers,

which contains a glide dislocation loop on an
inclined slip-plane. The solid is subjected to a
d
uniaxial tensile stress  0 perpendicular to the
layers.
5.9.9.1.
Calculate the total energy of the dislocation loop, in terms of the applied stress and relevant
geometric and material parameters. Use non-singular dislocation theory to calculate the
self-energy of the loop.
5.9.9.2.
Suppose that the layer contains a large number of dislocation loops with initial width d 0 .
The layer starts to deform plastically if the stress is large enough to cause the loops to
expand in the plane of the film (by increasing the loop dimension d). Calculate the yield
stress of the composite. How does the yield stress scale with film thickness?
76
5.10. Rayleigh-Ritz Method
5.10.1. Use the Rayleigh-Ritz method to obtain the natural frequency of
vibration of the spring-mass system shown (the displacement
associated with the vibration mode is trivial)
5.10.2. Use the Rayleigh-Ritz method to estimate
the fundamental frequency of the spring-mass
system shown. You should be able to obtain an
exact result, by describing the mode shape in
terms of a single parameter, and minimizing the
frequency appropriately.
k
k
m
k
m
m
5.10.3. Reconsider problem 5.10.2. Try to find the second frequency of vibration for the system by
selecting another approximation to the mode shape, which is (by construction) orthogonal to the first.
5.10.4. Use the Rayleigh-Ritz method to estimate the
fundamental frequency of the clamped-pinned beam
illustrated in the figure. Assume that the beam has Young’s
modulus E and mass density  , and its cross-section has
area A and moment of area I .
5.10.5. Use the Rayleigh-Ritz method to estimate the
fundamental frequency of the pinned-pinned beam illustrated
in the figure. Assume that the beam has Young’s modulus E
and mass density  , and its cross-section has area A and
moment of area I .
5.10.6. A beam with length L Young’s modulus E and mass
density  , and its cross-section has area A and moment of
area I is bonded to an elastic foundation, which exerts a
restoring force per unit length p   kw( x1 )e1 on the beam.
The beam is pinned at both ends. Use the Rayleigh-Ritz
method to estimate the natural frequency of vibration of the
beam.